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Node

A node (/noʊd/, from Latin nodus meaning "knot") is a point of connection, intersection, or joint. The term has multiple meanings across various fields of study and application, including mathematics, physical sciences, life sciences, technology and engineering, and music.^[1]

Mathematics

Graph theory

In graph theory, a node, also known as a vertex, is a fundamental element of a graph, representing one of the discrete points that may be connected to others via edges, which denote relationships or connections between them.^[2] Graphs are mathematical structures consisting of a set of vertices and a set of edges linking pairs of vertices, providing a framework for modeling pairwise relations in discrete mathematics.^[3] The origins of graph theory, and thus the concept of nodes, trace back to 1736 when Leonhard Euler addressed the Seven Bridges of Königsberg problem, modeling the city's landmasses as vertices and bridges as edges to determine if a closed walk traversing each bridge exactly once was possible.^[4] Euler's solution proved impossible due to the odd degrees of certain vertices, establishing key principles that formalized graph theory as a branch of mathematics.^[5] Vertices are classified by their degree, defined as the number of edges incident to them.^[6] An isolated vertex has degree zero, meaning it connects to no edges and stands alone in the graph.^[7] A pendant vertex, or leaf, has degree one, connecting to exactly one edge and often appearing at the ends of paths or branches.^[7] Internal vertices, typically with degree at least two, form the core structure of connected components, such as non-leaf nodes in trees.^[8] Key concepts involving nodes include adjacency, where two vertices are adjacent if an edge directly connects them, enabling analysis of local connectivity.^[9] Nodes are also represented in matrices: the adjacency matrix uses rows and columns labeled by vertices to indicate connections with 1s and 0s, while the incidence matrix has rows for vertices and columns for edges, marking 1 where a vertex meets an edge.^[9]^[10] In discrete mathematics, nodes underpin structures like paths (sequences of adjacent vertices), cycles (closed paths), and trees (acyclic connected graphs), facilitating problems such as route optimization where vertices model entities like cities and edges represent travel distances.^[11] For instance, finding the shortest path between two city vertices minimizes total edge weights, a foundational application in combinatorial optimization.^[12] These concepts extend briefly to directed graphs, where edges have orientation but vertices retain their role as connection points.^[3]

Geometry

In algebraic geometry, a node is a type of singularity on a curve where two branches intersect transversely, known as an ordinary double point. This occurs at a point where the curve equation f(x,y) = 0 satisfies f = f_x = f_y = 0, with the partial derivatives indicating distinct tangent lines along the branches.^[13] Locally, the singularity is analytically equivalent to the curve defined by xy = 0 in the completion of the local ring, representing two crossing lines.^[14] Examples of nodes appear in conic sections when the conic degenerates into two intersecting lines, forming a singularity at their intersection point. In elliptic curves, a node arises if the discriminant \Delta = 4A^3 + 27B^2 = 0 and the cubic polynomial x^3 + Ax + B has a double root, resulting in a singular curve where the branches cross at the singularity.^[15] Such nodes can be resolved through blow-up, a process that replaces the singular point with a projective line (the exceptional divisor), separating the branches and yielding a smooth surface; for a nodal cubic, a single blow-up suffices to achieve resolution.^[16] In topology, a node refers to a vertex in the structure of polyhedra, where edges meet at a point in the boundary complex. These nodes form the discrete points in the graph-theoretic representation of polyhedral surfaces, capturing the connectivity essential to topological invariants like Euler characteristic. In knot theory, nodes similarly denote the vertices of polygonal knot diagrams, serving as fixed points in the embedding where strands connect without self-intersection in the projection.^[17] This usage aligns analogously with vertices in graph theory but emphasizes continuous deformations in topological spaces. In spherical geometry, nodes denote the intersection points of great circles—the geodesics on the sphere—with reference planes passing through the center, such as the poles formed by the intersection of the polar axis with the equatorial plane. These points mark key locations where spherical coordinates align, facilitating measurements of angular distances along great circle arcs.^[18] The term "node" for such singularities in projective geometry traces back to Isaac Newton's 17th-century classification of cubic curves, where he identified and enumerated species featuring nodes as self-intersections, contributing to early understandings of curve singularities through projective transformations.

Analysis

In mathematical analysis, a node refers to a point where a function vanishes, equivalent to a root or zero of that function. For instance, the solutions to the trigonometric equation \sin x = 0 occur at nodes x = k\pi, where k is any integer, illustrating periodic zeros spaced by \pi.^[19] These nodes arise naturally in the study of oscillatory functions and form the basis for understanding more complex zero structures in analysis. In the context of differential equations, nodal points are the interior zeros of solutions to Sturm-Liouville boundary value problems, where the equation takes the form \frac{d}{dx} \left[ p(x) \frac{dy}{dx} \right] + [q(x) + \lambda w(x)] y = 0 with appropriate boundary conditions. A key property is the oscillation theorem, which states that the nth eigenfunction has exactly n-1 nodal points in the open interval, separating regions of sign change and ensuring orthogonality among eigenfunctions.^[20] This count of nodes distinguishes eigenfunctions by their order and is fundamental to spectral theory without relying on physical interpretations. Fourier analysis employs nodes in its harmonic basis functions, which serve as eigenfunctions for operators like the Laplacian on bounded domains. For the standard Fourier sine series on [0, \pi], the basis function \sin(nx) exhibits n-1 interior nodes at x = k\pi/n for k = 1, \dots, n-1, reflecting the harmonic structure of periodic expansions.^[21] In higher harmonics, these nodes capture the increasing oscillation frequency inherent to the series representation of square-integrable functions. Special functions in analysis, such as orthogonal polynomials and Bessel functions, feature nodes as their zeros, crucial for approximations and expansions. The Legendre polynomial P_n(x) of degree n has n real, distinct nodes in the interval (-1, 1), used as quadrature points in Gaussian integration schemes.^[22] Similarly, the Bessel function of the first kind J_\nu(z) possesses infinitely many positive real zeros, all simple except possibly at z=0, which act as nodes in series solutions to radial problems.^[23] For high-frequency or large-degree cases, such as the nth harmonic or high-order polynomials, the asymptotic spacing between consecutive nodes approximates \Delta x \approx \pi / n, reflecting denser oscillations in the bulk region.^[24]

Physical sciences

Astronomy

In astronomy, particularly celestial mechanics, an ascending node is defined as the point where an orbiting body crosses a reference plane, such as the ecliptic or equatorial plane, from south to north, while the descending node is the opposite crossing from north to south.^[25] These nodes are critical for describing the orientation of inclined orbits relative to the reference plane.^[26] The longitude of the ascending node, denoted by the symbol Ω, is a key orbital element that measures the angle from a fixed reference direction—typically the vernal equinox—to the ascending node, measured along the reference plane in the direction of orbital motion.^[27] This parameter, part of the six classical Keplerian orbital elements, is calculated using astrometric observations or numerical integration of perturbed trajectories, often via software like NASA's SPICE toolkit for precise ephemerides.^[28] For instance, in planetary orbits, Ω helps quantify the spatial alignment and is derived from position vectors at the node crossings. Nodes play a vital role in understanding orbital perturbations, where gravitational influences from other bodies cause shifts in the nodal positions, leading to nodal precession—the gradual rotation of the orbital plane around the system's angular momentum axis. In planetary systems, such perturbations can arise from oblateness of the central body or mutual interactions, altering long-term orbital stability. A prominent example is the Moon's nodal precession, driven primarily by the Sun's torque on Earth's equatorial bulge, completing a full cycle every 18.6 years and influencing eclipse seasons and tidal patterns.^[29] Historically, the concept of lunar nodes traces back to Babylonian astronomers around the 8th century BCE, who identified the draconic month—the 27.212-day interval between consecutive passages through the same node—through meticulous eclipse records, enabling predictions via cycles like the Saros.^[30] This empirical knowledge laid foundational insights into orbital inclinations without modern mathematics. In contemporary applications, orbital nodes are essential for satellite tracking, where the right ascension of the ascending node (a variant of Ω) defines ground track repeatability and collision avoidance in low-Earth orbits.^[25] Recent advancements in exoplanet studies, particularly from NASA's Transiting Exoplanet Survey Satellite (TESS) mission since 2018, have extended nodal analysis to detect precession in distant systems, revealing misaligned orbits that inform formation theories.

Physics

In wave physics, a node refers to a point along a standing wave where the displacement of the medium remains zero at all times, positioned between antinodes where the amplitude is maximum. This stationary behavior arises from the destructive interference of two waves of equal frequency and amplitude traveling in opposite directions, resulting in no net oscillation at the node. Such nodes are fundamental to understanding resonance and vibration in physical systems, including strings, air columns, and membranes.^[31]^[32] For a string fixed at both ends, standing waves form only at specific frequencies, with nodes always present at the fixed endpoints. The allowed wavelengths satisfy the relation \lambda = \frac{2L}{n}, where L is the length of the string and n = 1, 2, [3, \dots](/page/3_Dots) is the mode number, corresponding to the harmonic; here, the total number of nodes (including the two at the ends) is n + 1, while the number of antinodes equals n. In the fundamental mode (n=1), there are two nodes and one antinode, with the entire length accommodating half a wavelength; higher modes introduce additional internal nodes, shortening the wavelength and increasing the frequency proportionally. This quantization of modes ensures that energy is exchanged efficiently between the driving source and the string only at resonant frequencies.^[33]^[34] Historical experiments vividly illustrate these principles. In the 1860s, German physicist Franz Melde demonstrated standing waves on a tense string driven transversely or longitudinally by a tuning fork, observing nodes and antinodes that confirmed the wavelength-frequency relation and highlighted the role of tension in wave speed. Similarly, Ernst Chladni's early 19th-century experiments with vibrating plates, dusted with sand and excited by a bow, revealed nodal lines—curves of zero vibration—forming intricate patterns that delineate the boundaries between oscillating regions, providing a visual map of two-dimensional standing waves on plates. These patterns, known as Chladni figures, depend on the plate's shape, material, and excitation frequency, with sand accumulating along the nodal lines due to minimal disturbance.^[35]^[36]^[37] In systems of coupled harmonic oscillators, normal modes represent collective oscillations where all parts vibrate at a single frequency, featuring nodal points analogous to those in single standing waves. These nodes occur where the displacement amplitude is zero in the mode shape, dividing the system into segments oscillating out of phase; for instance, in a chain of masses connected by springs, the lowest-frequency mode has no internal nodes (all masses in phase), while higher modes introduce progressively more nodes, increasing the effective stiffness and frequency. This modal analysis is essential for predicting vibrational spectra in solids and molecules.^[38]^[39] Extending to quantum mechanics, nodes manifest as nodal surfaces in atomic and molecular wavefunctions, regions where the probability density of finding a particle is exactly zero due to the wavefunction changing sign. For hydrogen-like atoms, the number of radial nodal surfaces is n - l - 1 and the number of angular nodal surfaces is l, where n is the principal quantum number and l the azimuthal quantum number; p-orbitals (l=1) feature one angular nodal plane passing through the nucleus, separating lobes of opposite phase and prohibiting electron presence on that plane. These quantum nodes enforce orthogonality between wavefunctions of different states and influence chemical bonding by dictating forbidden regions for electron density.^[40]^[41] Recent advancements in quantum computing leverage these nodal structures in qubit wavefunctions, particularly through fixed-node approximations in quantum Monte Carlo simulations to mitigate the fermion sign problem and accurately model multi-electron systems on noisy intermediate-scale quantum hardware. As of 2023, such methods have enabled variational quantum algorithms to approximate ground-state energies with reduced bias, facilitating applications in quantum chemistry and materials design using platforms with tens of qubits.

Life sciences

Botany

In botany, a node is defined as the point along a plant stem where leaves, branches, aerial roots, or flowers attach, serving as a key structural and developmental junction. These nodes alternate with internodes, the elongated segments between them, and are essential for organizing plant architecture. Nodes occur along the stem, including near the tip, and are sites where leaves attach and axillary buds may emerge to form lateral branches. Nodes are particularly prominent in vascular plants, where they facilitate the emergence of lateral structures from meristematic tissues. The arrangement of leaves emerging from successive nodes, known as phyllotaxy, influences light capture and overall plant efficiency; common patterns include alternate (one leaf per node in a spiral), opposite (two leaves per node directly across from each other), and whorled (three or more leaves per node in a circle). This phyllotactic diversity optimizes photosynthetic exposure and is governed by hormonal signals and genetic factors during development. Illustrative examples of nodes appear in various plant groups, such as grasses (Poaceae family), where swollen nodes at the base produce tillers—lateral shoots that enable vegetative propagation and dense growth in pastures. In climbing vines like ivy (Hedera spp.), nodes often generate adventitious roots for attachment to supports. Internode lengths between nodes are a critical trait in crop breeding; for instance, shorter internodes in maize (Zea mays) and soybean (Glycine max) varieties reduce plant height, improving resistance to lodging and yield stability under high-density planting. Biologically, nodes are hotspots of meristematic activity, housing undifferentiated cells in axillary meristems that promote branching, regeneration after injury, and adaptation to environmental stresses. This regenerative capacity allows plants to repair damage or produce new shoots from nodal buds, supporting survival and propagation. The term "node," derived from the Latin nodus meaning "knot," entered botanical nomenclature in the 18th century during the Linnaean era, when Carl Linnaeus formalized plant descriptions in works like Species Plantarum (1753), using it to denote stem joints in taxonomic classifications. In modern genetics, node formation in the model plant Arabidopsis thaliana is regulated by key genes such as LATERAL SUPPRESSOR (LAS), which initiates axillary meristems at nodes to control branching patterns, as identified in seminal studies on shoot architecture.

Medicine

In medicine, nodes primarily refer to lymph nodes, which are small, bean-shaped structures integral to the immune system. These nodes, numbering approximately 600 in the human body, are distributed along lymphatic vessels and serve as filters for lymph fluid, trapping bacteria, viruses, and other foreign particles while housing lymphocytes that initiate immune responses.^[42]^[43] Each node consists of a cortex rich in B cells and a paracortex dominated by T cells, enabling antigen presentation and antibody production to combat infections.^[42] Other notable nodal structures include sentinel lymph nodes and the tonsils. Sentinel lymph nodes are the initial drainage points from a primary tumor, playing a critical role in cancer staging by indicating whether metastasis has occurred; biopsy of these nodes helps determine the need for further lymph node dissection in cancers like breast and melanoma.^[44] The tonsils, which form part of Waldeyer's ring in the pharynx, are dense lymphoid aggregates embedded in mucosal tissue that protect against ingested or inhaled pathogens, functioning similarly to peripheral lymph nodes.^[45]^[46] Pathological conditions often manifest as nodal enlargement, known as lymphadenopathy, which can signal infection or malignancy. For instance, infectious mononucleosis, caused by the Epstein-Barr virus, commonly leads to swollen cervical and axillary lymph nodes due to immune activation, accompanied by symptoms like fever and sore throat.^[47]^[48] Diagnostic evaluation typically involves biopsy techniques such as fine-needle aspiration for cytological analysis, core needle biopsy for histological detail, or excisional biopsy to remove an entire node when malignancy is suspected.^[49]^[50] Historically, the lymphatic system's nodes were first systematically identified in the 17th century through vivisections revealing chyle vessels and glands, building on ancient descriptions of glandular swellings.^[51] Contemporary diagnostics rely on advanced imaging, such as positron emission tomography (PET) scans, which detect metabolic activity in nodes to identify metastasis in cancers like cervical and head-and-neck tumors with high sensitivity.^[52] Recent progress includes post-2022 clinical trials investigating immunotherapies that target tumor-draining lymph nodes to boost T-cell responses, such as phase I studies extracting and reinfusing node-derived T cells in solid tumors.^[53]^[54]

Technology and engineering

Computing

In computing, a node serves as a fundamental building block in various data structures, representing discrete units of data connected through links or pointers. In linked lists, each node typically contains data and a reference to the next node, enabling dynamic memory allocation and efficient insertions or deletions without shifting elements. Trees extend this concept hierarchically, where nodes are organized with a root node at the top and child nodes branching downward, facilitating operations like searching and sorting in balanced structures such as binary search trees. Abstract syntax trees (ASTs) apply nodes to represent the syntactic structure of source code in compilers, with each node denoting operators, variables, or expressions to enable parsing and optimization. Node.js emerged as a prominent server-side JavaScript runtime environment, first released in 2009 by Ryan Dahl to address limitations in traditional web servers like Apache by leveraging Google's V8 engine. Its event-driven, non-blocking I/O architecture allows for handling multiple concurrent connections efficiently on a single thread, making it suitable for scalable network applications such as real-time web services. This design draws from graph theory's vertex model, where nodes interact asynchronously via callbacks or promises. Traversal algorithms operate on these node-based structures to explore and process data systematically. Depth-First Search (DFS) begins at the root node and delves deeply into one branch before backtracking, often implemented recursively and ideal for pathfinding in trees. Breadth-First Search (BFS), in contrast, explores all nodes at the current depth level before proceeding deeper, using a queue to visit siblings level by level from the root, which is efficient for shortest-path problems in unweighted trees. In web development, Document Object Model (DOM) nodes form a tree representation of HTML or XML documents, allowing JavaScript to dynamically manipulate page elements like text or attributes. Blockchain systems employ nodes for distributed ledger maintenance, where full nodes store the entire chain and independently validate all transactions against consensus rules, ensuring network integrity, while light nodes query full nodes for headers only, reducing storage needs at the cost of full verification autonomy. Serverless computing has evolved to integrate Node.js deeply, with platforms like AWS Lambda advancing in 2024 by supporting Node.js 22 runtime and enhanced container images for faster cold starts and better performance in event-driven workloads. These updates enable developers to deploy Node.js functions without managing infrastructure, scaling automatically for microservices and APIs, as seen in Lambda's expanded support for Python and Node.js versions to handle diverse serverless architectures.

Networks and electronics

In computer networks, a node is defined as any physical device connected to a network that can send, receive, or forward data, including examples such as routers, sensors, computers, printers, and servers. These nodes operate within local area networks (LANs) or wide area networks (WANs), where they communicate using standardized protocols to exchange information. Each node is assigned a unique Internet Protocol (IP) address, a 32-bit identifier that combines a network prefix to specify the subnet and a host portion to identify the specific device, enabling efficient packet routing across interconnected systems.^[55]^[56] In electrical and electronic circuits, a node represents a junction or connection point where two or more circuit elements, such as resistors, capacitors, or wires, meet, allowing current to flow between components. This concept is fundamental to circuit analysis through Kirchhoff's current law (KCL), which asserts that the algebraic sum of all currents entering and exiting a node equals zero, reflecting the principle of charge conservation at the junction. KCL applies to both direct current (DC) and alternating current (AC) circuits, providing a basis for solving complex networks by balancing inflows and outflows without net accumulation of charge.^[57] Wireless sensor networks (WSNs) in Internet of Things (IoT) applications rely on densely deployed nodes—typically low-power sensors and actuators—that collect environmental data and transmit it via short-range wireless protocols. The Zigbee protocol, built on the IEEE 802.15.4 standard, facilitates this by organizing nodes into a mesh topology with three primary types: coordinators that initialize and manage the network, routers that forward data across multiple hops to extend coverage, and end devices that perform sensing with minimal energy use. This structure supports scalable IoT deployments, such as smart homes or industrial monitoring, by enabling self-healing communication paths and low-latency data aggregation while conserving battery life in remote nodes.^[58]^[59] Mesh networks exemplify node-based routing in practice, where intermediate routing nodes dynamically relay packets between source and destination, creating redundant paths to improve fault tolerance and coverage in environments like urban IoT or ad-hoc systems. Historically, the ARPANET—the foundational network for the modern internet—launched in 1969 with four initial nodes at the University of California, Los Angeles (UCLA), Stanford Research Institute, University of California, Santa Barbara, and the University of Utah, using interface message processors to enable packet-switched communication across these sites. As of 2025, advancements in 5G networks have integrated edge nodes within multiaccess edge computing (MEC) frameworks, deploying high-performance servers at network peripheries to process IoT data with ultra-low latency, supporting applications like autonomous vehicles and real-time analytics; 6G research extends this with AI-driven edge intelligence for even greater scalability and predictive routing.^[60]^[61]^[62] Software nodes, as logical representations, mirror these physical counterparts in simulations for testing network behaviors.

Aerospace

In aerospace engineering, node modules serve as critical connecting elements in space stations and spacecraft, facilitating the integration of multiple components while providing pressurized habitable volumes and docking capabilities. These structures enable modular assembly in orbit, allowing for expansion, maintenance, and multi-vehicle docking. The concept traces its origins to early space station designs, where nodes act as central hubs analogous to network junctions, linking habitats, laboratories, and support systems.^[63] The first implementation of a node-like module occurred with NASA's Skylab space station, launched on May 14, 1973, aboard a Saturn V rocket. Skylab's Multiple Docking Adapter (MDA), a pressurized compartment measuring approximately 5.2 meters in length and 3.2 meters in diameter, featured two forward docking ports and an observation window cluster, enabling multiple spacecraft to attach simultaneously for crew rotations and resupply. This approximately 4.3-ton component, integrated with the airlock module, provided a pressurized volume of 33 cubic meters and served as a precursor to modern nodes by supporting extravehicular activities and telescope operations.^[64]^[65]^[66] Subsequent advancements materialized in the International Space Station (ISS), where node modules became foundational. Unity (Node 1), launched on December 4, 1998, aboard Space Shuttle Endeavour during STS-88, was the first U.S.-built ISS component—a cylindrical aluminum structure 5.49 meters long and 4.57 meters in diameter, offering a pressurized volume of 75.1 cubic meters. Equipped with six Common Berthing Mechanism (CBM) ports (two axial and four radial), it connected the Russian Zarya module to future elements like the Destiny laboratory, enabling orbital assembly. Later nodes, such as Harmony (Node 2, launched 2007) and Tranquility (Node 3, launched 2010), expanded this design with similar multi-port configurations for enhanced connectivity and internal storage. In rocketry contexts, these nodes support docking for orbital assembly, as seen in the ISS's progressive buildup using shuttle and automated vehicles.^[67]^[68]^[69] Node designs prioritize structural integrity, with pressurized shells constructed from high-strength aluminum alloys to withstand launch loads and maintain internal atmospheres at 101.3 kPa. Multiple ports accommodate radial and axial attachments via standardized adapters, facilitating crew transfer and equipment exchange without depressurization. Radiation shielding is incorporated through the module's multi-layer insulation, aluminum walls (providing ~2-5 g/cm² areal density), and optional polyethylene liners, reducing galactic cosmic ray exposure by up to 30% compared to unshielded areas, though supplemental vests are used for high-risk tasks. These features ensure habitability during long-duration missions.^[70]^[71] Contemporary developments extend node concepts to deep space. In NASA's Artemis program, the Lunar Gateway—a cislunar outpost—incorporates connecting elements like the Habitation and Logistics Outpost (HALO) module, with the module arriving in the United States in April 2025 for final integration, planned for launch no earlier than 2027. It features multiple docking ports for Orion spacecraft and lunar landers, providing a pressurized volume of ~50 cubic meters with integrated shielding against solar particle events. Private initiatives, such as Axiom Space's Axiom Station, with its primary structure for the first module relocated to Houston no earlier than fall 2025, aim to attach initial modules to the ISS no earlier than 2027, evolving into a free-flying station by 2030 with dedicated docking nodes for commercial vehicles, emphasizing reusable ports and enhanced radiation protection via advanced composites. These nodes underscore the shift toward scalable, international orbital infrastructure.^[72]^[73]^[74]

Music

Acoustics

In stationary sound waves, a node refers to a point of zero pressure amplitude, where the acoustic pressure does not deviate from the ambient atmospheric pressure. This condition represents minimum pressure variation and is analogous to a node in transverse standing waves, where the transverse displacement is zero. In longitudinal sound waves, pressure nodes coincide with displacement antinodes, locations of maximum particle motion along the propagation direction.^[75] Resonance tubes, such as those in organ pipes, exhibit standing sound waves governed by boundary conditions at the ends. In a pipe closed at one end and open at the other, the closed end forms a displacement node—where air particle displacement is zero—and a pressure antinode, with maximum pressure variation due to the rigid boundary preventing motion. Conversely, the open end is a displacement antinode and a pressure node, as the pressure remains constant at atmospheric levels while allowing free particle oscillation. These configurations ensure reflection and interference produce stable standing patterns.^[76]^[77] The frequencies of resonant modes depend on the number and positions of nodes. For the fundamental mode in a closed pipe, there is one pressure node at the open end (or one displacement node at the closed end), corresponding to a quarter-wavelength fitting the pipe length L = \lambda/4, yielding the fundamental frequency f_1 = v/(4L), where v is the speed of sound. Overtones occur at odd harmonics (f_n = (2n-1) f_1, n = 1, 2, 3, \dots), adding internal nodes; for instance, the first overtone (third harmonic) includes one additional pressure node inside the tube alongside the end node, resulting in L = 3\lambda/4. This node progression determines the harmonic series unique to closed pipes.^[76] Kundt's tube experiment visualizes standing sound waves to measure the speed of sound by identifying displacement node positions. A sound source generates longitudinal waves in a tube partially filled with fine powder like lycopodium; the powder accumulates at displacement nodes, where minimal air motion occurs, forming visible piles separated by \lambda/2. Measuring this spacing d between consecutive nodes and the driving frequency f allows calculation of v = 2 f d, providing empirical verification of sound speed in air, typically around 343 m/s at room temperature.^[78]^[79] Mathematically, nodes arise from solutions to the acoustic wave equation \frac{\partial^2 p}{\partial t^2} = v^2 \frac{\partial^2 p}{\partial x^2}, where p(x,t) is the pressure deviation from equilibrium. Separable solutions yield standing waves of the form p(x,t) = \psi(x) \cos(\omega t), with the spatial function \psi(x) satisfying \psi(x) = 0 at pressure nodes. For a closed-open pipe, \psi(x) = A \cos(kx) with k = (2n-1)\pi / (2L), ensuring \psi(0) = A (antinode at closed end x=0) and \psi(L) = 0 (node at open end x=L). Higher modes increase n, introducing additional zeros in \psi(x).^[76]

Instruments

In musical instruments, nodes refer to the points along a vibrating medium—such as a string, air column, or membrane—where the displacement amplitude is zero in a standing wave pattern. These stationary points arise from the interference of waves traveling in opposite directions, creating resonant modes that determine the instrument's pitch and timbre. The positions of nodes, alternating with antinodes (points of maximum displacement), are fixed by the instrument's geometry and boundary conditions, enabling the production of fundamental tones and harmonics.^[80] In string instruments like the guitar, violin, and piano, the strings are fixed at both ends, establishing nodes at the bridge and nut (or tuning pegs). This configuration supports standing transverse waves where the fundamental mode has one antinode in the center, with wavelength \lambda_1 = 2L (where L is the string length) and frequency f_1 = \frac{v}{2L}, with v = \sqrt{\frac{F}{\mu}} being the wave speed determined by tension F and linear density \mu. Higher harmonics occur when additional nodes divide the string into equal segments, yielding frequencies f_n = n f_1 for integer n, producing overtones that enrich the sound; for instance, lightly touching a guitar string at its midpoint isolates the second harmonic (an octave above the fundamental). The sounding board or body amplifies these vibrations, but the nodal pattern on the string governs the core acoustics.^[80]^[81] Wind instruments, including flutes, clarinets, and trumpets, rely on standing longitudinal pressure waves in air columns, where nodes represent points of minimal pressure variation (typically at open ends) and antinodes occur at maximum pressure changes (at closed ends or reeds). For open pipes like the flute, both ends are effectively open, placing nodes there and allowing all harmonics f_n = n \frac{v}{2L} (with v \approx 343 m/s at room temperature), though finger holes adjust effective length L to select modes. In closed pipes such as the clarinet, a node forms at the open end and an antinode at the reed end, restricting resonances to odd harmonics f_n = (2n-1) \frac{v}{4L}; for example, a standard Bb clarinet has an approximate length of 0.66 m and a fundamental frequency of about 147 Hz (sounding D3), with end corrections increasing the effective frequency from the uncorrected value of around 130 Hz.^[81]^[82] Brass instruments like the trumpet operate similarly but with a narrower bore and lip vibration exciting primarily odd harmonics, contributing to their bright timbre. End corrections (adding about 0.6 times the radius to L) refine these calculations for accuracy.^[81] Percussion instruments, such as drums and cymbals, involve more complex nodal patterns on membranes or plates, where circular or radial nodes form standing waves from initial strikes. In a drumhead, the fundamental mode features a nodal circle near the edge, with overtones from higher radial or angular nodes producing inharmonic spectra that give distinctive timbres. These nodal structures, analyzed through Bessel functions in advanced acoustics, differ from the linear patterns in strings and winds but similarly dictate resonant frequencies.^[81]

References

[1]
About Node.js
Node.js® is a free, open-source, cross-platform JavaScript runtime environment that lets developers create servers, web apps, command line tools and ...
[2]
Why the creator of Node.js® created a new JavaScript runtime
Mar 19, 2024 · Ryan Dahl, creator of Node.js and Deno, tells us about his journey into software development and the creation of Node.js. He explains why he ...
[3]
History of Node.js on a Timeline - RisingStack Engineering
May 29, 2024 · Node.js was named in 2009, had early npm preview, and saw mainstream adoption in 2017, with 8.8 million online instances.Node.js milestones · Node.js in 2017 – the year of... · Node.js in 2018 – Node 10...
[4]
What Is Node.js and Why You Should Use It
Oct 1, 2025 · Node.js is a single-threaded, open-source, cross-platform runtime environment for building fast and scalable server-side and networking ...
[5]
What is the Node.js (Node) runtime environment? - TechTarget
Nov 30, 2022 · Node.js (Node) is an Open Source, cross-platform runtime environment for executing JavaScript code. Node is used extensively for server-side programming.
[6]
What exactly is Node.js? - freeCodeCamp
Apr 18, 2018 · Here's a formal definition as given on the official Node.js website: Node.js® is a JavaScript runtime built on Chrome's V8 JavaScript engine.
[7]
Graph Vertex -- from Wolfram MathWorld
"Vertex" is a synonym for a node of a graph, i.e., one of the points on which the graph is defined and which may be connected by graph edges.
[8]
Graph -- from Wolfram MathWorld
The points of a graph are most commonly known as graph vertices, but may also be called "nodes" or simply "points." Similarly, the lines connecting the vertices ...
[9]
[PDF] Early Writings on Graph Theory: Euler Circuits and The Königsberg ...
Dec 8, 2005 · In it, Euler undertakes a mathematical formulation of the now-famous Königsberg Bridge Problem: is it possible to plan a stroll through the town.
[10]
[PDF] The Origins of Graph Theory 1 Two problems - Jeremy Martin
2 Graph theory. In 1736, the great Swiss mathematician Leonhard Euler solved the Königsberg bridge problem. Euler's key insight was that the islands and ...
[11]
Vertex Degree -- from Wolfram MathWorld
The degree of a graph vertex of a graph , also called the vertex degree or local degree, is the number of graph edges which touch.
[12]
Pendent Vertex, Isolated Vertex and Adjacency of a graph
Aug 23, 2019 · By using degree of a vertex, we have a two special types of vertices. A vertex with degree one is called a pendent vertex.
[13]
[PDF] Elements of Graph Theory
A leaf node in a tree or forest is any pendant or isolated vertex. An internal node is any non-leaf vertex (thus it has degree ≥). Page 50. 50. Trees as ...
[14]
Adjacency Matrix -- from Wolfram MathWorld
The adjacency matrix, sometimes also called the connection matrix, of a simple labeled graph is a matrix with rows and columns labeled by graph vertices.
[15]
Incidence Matrix -- from Wolfram MathWorld
The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge ...
[16]
[PDF] GRAPH THEORY WITH APPLICATIONS
This book is intended as an introduction to graph theory. Our aim has been to present what we consider to be the basic material, together with a wide.
[17]
[PDF] Computing the Shortest Path: A∗ Search Meets Graph Theory
We study the problem of finding a shortest path between two vertices in a directed graph. This is an important problem with many applications, ...
[18]
Ordinary Double Point -- from Wolfram MathWorld
An ordinary double point of a plane curve is point where a curve intersects itself such that two branches of the curve have distinct tangent lines.
[19]
Section 53.19 (0C46): Nodal curves—The Stacks project
We say a closed point x \in X is a node, or an ordinary double point, or defines a nodal singularity if there exists an ordinary double point \overline{x} \in X ...
[20]
Nodal and cuspidal curves - Purdue Math
The degenerate conic consisting of two nonparallel lines has a singularity where they meet. It is easy to construct singular cubics with several components ...Missing: sections | Show results with:sections
[21]
[PDF] Blow-ups
Thus nodes and cusps are resolved by one blow up and the formal genera is reduced by one. Hence a cubic with a node or a cusp becomes a P1 after one blow up.
[22]
[PDF] Polyhedra and Euler Characteristics - MIT Mathematics
The polyhedron's vertices are represented as nodes, and the edges of the polyhedron become edges of the graph. This method enables the formation of a ...
[23]
Great Circle -- from Wolfram MathWorld
A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a ...
[24]
Graphs of trigonometric functions - The Math Page
The graph of y = sin x. The zeros of y = sin x are at the multiples of π. It is there that the graph crosses the x-axis, because there y = 0. And its period ...Missing: nodes | Show results with:nodes
[25]
Traces and inverse nodal problem for Sturm–Liouville operators with ...
Inverse nodal problems consist in recovering operators from zeros of their eigenfunctions, called nodal points.
[26]
Differential Equations - Fourier Series - Pauls Online Math Notes
Nov 16, 2022 · In this section we define the Fourier Series, i.e. representing a function with a series in the form Sum( A_n cos(n pi x / L) ) from n=0 to ...
[27]
18.38 Mathematical Applications
Classical OP's play a fundamental role in Gaussian quadrature. If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros ...
[28]
Bessel Function Zeros -- from Wolfram MathWorld
The functions J_nu(z), J_nu^'(z), Y_nu(z), and Y_nu^'(z) each have an infinite number of real zeros, all of which are simple with the possible exception of z=0.
[29]
[PDF] Asymptotics of orthogonal polynomials and the Painlevé ... - arXiv
Sep 14, 2016 · In this article, we will review asymptotic results of some orthogonal polynomials which are related to the. Painlevé equations. However, due to ...
[30]
Chapter 5: Planetary Orbits - NASA Science
Nodes are points where an orbit crosses a reference plane, such as the ecliptic or the celestial equator. As an orbiting body crosses the reference plane going ...
[31]
Frequently Asked Questions (FAQs) - JPL Solar System Dynamics
... (ω) is the argument of perihelion, and upper case omega (Ω) is the longitude of the ascending node. The symbol to the far left (the rams horn - symbol of ...
[32]
Approximate Positions of the Planets - JPL Solar System Dynamics
longitude of perihelion [degrees, degrees/century]. Ω o , Ω ˙, longitude of the ascending node [degrees, degrees/century]. In order to obtain the coordinates ...
[33]
Planetary Satellite Mean Elements - JPL Solar System Dynamics
Mean orbital parameters are primarily useful in describing the general shape and orientation of a planetary satellite's orbit.
[34]
NASA: Moon Movements to Amplify Sea-Level Rise, Flooding in Mid ...
Aug 4, 2021 · A full lunar nodal precession cycle takes about 18.6 years to complete, marked by two distinct periods. In the first half, when the moon is ...
[35]
NASA - Eclipses and the Saros
Jan 12, 2012 · One Saros is equal to 223 synodic months. However, 239 anomalistic months and 242 draconic months are also equal to this same period (to within ...
[36]
[PDF] Is the orbit of the exoplanet WASP-43b really decaying? TESS and ...
Oct 11, 2021 · This interaction has various forms: apsidal precession in which the orbit ellipse rotates in its own plane, and nodal precession in which the.
[37]
The TESS-Keck Survey. IV. A Retrograde, Polar Orbit for the Ultra ...
We find that nodal precession and disk dispersal-driven tilting can both explain the current orbital geometry while Kozai-Lidov cycles are suppressed by general ...
[38]
Standing waves
All standing waves are characterized by positions along the medium which are standing still. Such positions are referred to as nodes.Missing: physics | Show results with:physics
[39]
16.6: Standing Waves and Resonance - Maricopa Open Digital Press
A node occurs at each end of the string. The nodes are boundary conditions that limit the possible frequencies that excite standing waves. (Note that the ...
[40]
Lab 1: Standing Waves
Fundamental: L = λ/2, n = 1, 1/2 wavelength fits into the length of the string. Second harmonic: L = λ n = 2, one wavelength fits into the length of the string.
[41]
Standing Waves: Explanation and Examples | Albert Blog & Resources
Sep 26, 2023 · For higher harmonics, the relationship can be generalized as λ = 2 n L \lambda = \frac{2}{n} L λ=n2L. By applying this relationship, ...
[42]
(PDF) Standing Waves in a Thin String (Melde's experiment)
Jan 12, 2019 · Melde's experiment is a scientific experiment carried out in 1859 by the German physicist Franz Melde on the standing waves produced in a tense cable.
[43]
44.45 -- Chladni plates - UCSB Physics
In a violin, the top is fixed to the side (or rib), which produces a node a small distance from the edge of the plate all around it. The edge of the violin- ...
[44]
Chladni Plates | Harvard Natural Sciences Lecture Demonstrations
When the plate is oscillating in a particular mode of vibration, the nodes and antinodes that are set up form complex but symmetrical patterns over its surface.
[45]
8.4: Coupled Oscillators and Normal Modes - Physics LibreTexts
Nov 8, 2022 · We move from a single oscillator into combinations of oscillators. A very basic first step is to consider the effect of multiple springs on a single mass.
[46]
[PDF] Chapter 3 - Normal Modes - MIT OpenCourseWare
All the normal modes have real ω2 . Thus there are only three possibilities. ω2 > 0 corresponds to stable equilibrium and harmonic oscillation. ω2 < 0, in which ...
[47]
2.2.2: Quantum Numbers and Atomic Wave Functions
Jan 31, 2024 · The wavefunction describes what we know as an atomic orbital; it defines the region in space where the electron is located. Additionally, there ...
[48]
5.1.5: Quantum Mechanics and Atomic Orbitals - Chemistry LibreTexts
Oct 2, 2025 · Nodal surfaces are directly connected with the energy of an orbital. The higher the energy of the orbital, the larger the number of nodal ...
[49]
Anatomy, Lymph Nodes - StatPearls - NCBI Bookshelf - NIH
Jul 25, 2023 · The primary function of lymph nodes is filtering interstitial fluid ... 8. Cupedo T. Human lymph node development: An inflammatory interaction.Introduction · Structure and Function · Embryology · Blood Supply and Lymphatics
[50]
Lymphatic Anatomy and Physiology - PMC - PubMed Central - NIH
Lymph nodes generally receive multiple afferent vessels and fill several roles, the most studied being immunologic function. The node itself houses lymphocytes ...
[51]
Sentinel Lymph Node Biopsy - NCI - National Cancer Institute
Jun 25, 2019 · SLNB is most commonly used to help stage breast cancer and melanoma. It is sometimes used to stage penile cancer (1) and endometrial cancer (2).
[52]
https://pmc.ncbi.nlm.nih.gov/articles/PMC8888156/
[53]
Tonsils: Anatomy, Definition & Function - Cleveland Clinic
Your tonsils are two round, fleshy masses in the back of your throat (pharynx). Part of your immune system, your tonsils are like lymph nodes.
[54]
Mononucleosis - Symptoms & causes - Mayo Clinic
Infectious mononucleosis (mono) is often called the kissing disease. ... But fatigue, enlarged lymph nodes and a swollen spleen may last for a few weeks longer.Mononucleosis: Can it recur? · Mononucleosis and Epstein-Barr
[55]
About Infectious Mononucleosis (Mono) | EBV and Mono - CDC
May 9, 2024 · Swollen lymph nodes in the neck and armpits; Swollen liver or spleen or both; Rash. Enlarged spleen and a swollen liver are less common symptoms ...
[56]
Lymph Node Biopsy: Procedure & Results - Cleveland Clinic
Apr 2, 2024 · During the procedure, a provider locates the lymph nodes closest to the tumor (sentinel lymph nodes) and removes them to test for cancer cells.
[57]
Types of Biopsies - Lymphoma Research Foundation
In this procedure, a surgeon cuts through the skin to remove an entire lymph node (excisional biopsy) or a large portion of tissue (incisional biopsy).
[58]
The discovery of the lymphatic system in the seventeenth century ...
In the seventeenth century, opportunities to discover chyle came about through the revival of vivisection. Gaspare Aselli discovered chyle vessels in a ...
[59]
New PET/CT criterion for predicting lymph node metastasis in ... - NIH
Jan 20, 2022 · The conventional criterion used for determining lymph node metastasis on PET/CT is a SUVmax of 2.5 or larger on axial images. , , Expert ...
[60]
Phase I Clinical Study of Tumor-associated Lymph Node T Cell ...
This is an open, single-center Phase I clinical trial designed to assess the safety, tolerability, efficacy, and feasibility of tumor-associated lymph node T ...
[61]
Immune microenvironment of tumor-draining lymph nodes
Apr 11, 2025 · Tumor-draining lymph nodes (TDLNs) play a crucial role in modulating tumor immune responses and influencing the efficacy of immunotherapy.
[62]
[PDF] Fundamentals of Computer Networks
– Nodes: PC, workstations, special-purpose hardware… • hosts. • switches. – Links: coax cable, optical fiber, wireless… • ...<|separator|>
[63]
[PDF] IP Addressing
An IP address: - is a 32 bit long identifier. - encodes a network number (network prefix) and a host number. Page 18. • The network prefix identifies a network ...
[64]
Kirchhoff's Current Law (KCL) - All About Circuits
Kirchhoff's Current Law, often shortened to KCL, states that “The algebraic sum of all currents entering and exiting a node must equal zero.”
[65]
What is Zigbee? Learn About Zigbee Wireless Mesh Technology
The Zigbee IoT protocol allows devices to communicate data seamlessly in a variety of network topologies using extremely limited power. Thanks to Zigbee, ...Advantages Of Zigbee 3.0... · Common Wireless Applications... · Zigbee Technology Faqs
[66]
(PDF) WIRELESS SENSOR NETWORK USING ZIGBEE
Aug 6, 2025 · WSN is composed of ZigBee coordinator (network coordinator), ZigBee router and ZigBee end device. The sensor nodes information in the network ...
[67]
What is a Mesh Network? | Soracom
A mesh network is a decentralized communication structure where multiple devices, or nodes, work together to route data from one point to another. Each node can ...
[68]
ARPANET | DARPA
1969. First four nodes of the ARPANET. University of California, Santa Barbara; University of California, Los Angeles; University of Utah; Stanford Research ...
[69]
https://www.nasa.gov/reference/international-space-station/
[70]
Skylab: America's First Space Station - NASA
May 14, 2018 · Skylab was America's first space station and first crewed research laboratory in space. Early visions of orbiting space stations predated the Space Age.Missing: node | Show results with:node
[71]
[PDF] 19740020218.pdf - NASA Technical Reports Server
... Skylab Orbiting Assembly l-h. 1.2.1-2. Skylab-. Function of Modules. 1-5. _ i. 2.1-3. Orbital Workshop. 1-7. : 1.2.1-4. Mission Design Profile. 1-8. 1.2.2.5-1.
[72]
50 Years Ago: The Launch of Skylab, America's First Space Station
May 14, 2023 · Skylab, America's first space station and the first crewed research laboratory in space, lifted off on May 14, 1973, on the last Saturn V rocket.Missing: node | Show results with:node
[73]
STS-88 - NASA
The STS-88 mission launched aboard the Space Shuttle Endeavour on December 4, 1998 at 2:35 a.m. (CST) to deliver the Unity Node to the International Space ...
[74]
Unity Module - NASA
Unity was the second module of the International Space Station. It was delivered to orbit aboard space shuttle Endeavour in December of 1998.
[75]
International Space Station - NASA
Jan 17, 2025 · In Phase 2, the elements of the new ISS were launched beginning in 1998. ... Node 1 Unity. Over the course of the first five shuttle assembly ...
[76]
[PDF] INTERNATIONAL SPACE STATION - NASA
The International Space Station (ISS) is the unique blend of unified and diversified goals among the world's space agencies that will lead to improvements in ...
[77]
[PDF] ISS Radiation Shielding and Acoustic Simulation Using an ...
The ISS Environment Simulator (ISSES) is an immersive application that allows the user to analyze, evaluate, and change the radiation and noise environments of ...
[78]
NASA Welcomes Gateway Lunar Space Station's HALO Module to US
Apr 4, 2025 · Gateway's HALO will provide Artemis astronauts with space to live, work, conduct scientific research, and prepare for missions to the lunar surface.
[79]
Axiom Space revises space station assembly plans - SpaceNews
Dec 18, 2024 · Axiom Space is revising the assembly sequence for its commercial space station, a move it says will allow it to get to a free-flying station sooner.
[80]
Standing Waves in Air Columns - oPhysics
When a standing wave forms in an air column, there is always a displacement antinode at any open end and a displacement node at any closed end. Note: ...
[81]
8.8: Standing Waves
### Summary of Standing Sound Waves
[82]
Standing Waves in Pipes
The closed end of the pipe is thus a displacement node. In order not to displace air the closed pipe end has to exert a force on the molecules by means of ...
[83]
Longitudinal Waves - Kundt's Tube - HyperPhysics
In this experiment, longitudinal sound waves will be produced in a metal rod and an air column. Using the properties of wave motion, the frequency of the sound
[84]
Kundt's Tube - UCLA Physics & Astronomy
Hot wire probe (designed by S. Baker) detects the position of the velocity antinodes (which are at the position of the pressure nodes). The observed frequency ...
[85]
Strings, standing waves and harmonics - UNSW
These motionless points are called nodes of the vibration, and they play an important role in nearly all of the instrument families. Halfway between the nodes ...
[86]
17.5 Sources of Musical Sound - University Physics Volume 1
Sep 19, 2016 · The antinodes do not occur at the opening of the tube, but rather depend on the radius of the tube. The waves do not fully expand until they ...