Wave Motion Equation Speed & Types Wave Motion Equation Speed & Types Waves move patterns, not matter. When you pluck a string, flick a rope, or speak, what travels from one place to another is a repeating disturbance in space and time. Nearby particles of the medium oscillate about their equilibrium positions, and this coordinated dancing passes the message along. The shape of this motion is described by a compact “progressive wave” equation that tells you the displacement of the medium at any position x and time t . From that one description you can read the amplitude (how big the oscillations are), the wavelength (how far until the pattern repeats in space), the frequency (how many repeats per second), and the speed (how fast the pattern moves). Crucially, wave speed is not set by how vigorously you shake (amplitude) but by the properties of the medium: tension and mass per length for a string; elasticity and density for sound in air, water, or steel. In everyday terms, a tight, light string transmits waves quickly; a slack, heavy string transmits them slowly. Similarly, sound travels faster in steel than in air because steel is far harder to compress. This lesson builds the progressive wave equation step by step, fixes sign conventions so you always know which way the wave travels, and then derives formulas connecting wave speed to medium parameters: v= T/ for strings, v= P/ for gases, and v= B/ for general fluids/solids. Along the way, you will see how v=f locks frequency and wavelength together: in a given medium where v is fixed, raising f makes smaller in exact proportion. We also organize wave types (transverse vs longitudinal, mechanical vs electromagnetic) and introduce wavefronts for a clean geometric picture. With this foundation, you can confidently solve NEET problems about speeds, harmonics, and identifying wave direction from an equation. remember A stadium wave looks like people are moving around the arena, but each person just stands up and sits down. Likewise, on a rope or in air, particles oscillate locally while the wave pattern travels. Wave A traveling disturbance that transfers energy and momentum without net transport of matter. The material through which a mechanical wave travels (string, air, water, steel). Electromagnetic waves do not require a medium. Medium Transverse Wave Particle oscillations are perpendicular to the direction of propagation (e.g., waves on a string, electromagnetic waves). Longitudinal Wave Particle oscillations are parallel to the direction of propagation (e.g., sound waves in air). Amplitude ( A ) Maximum displacement of a particle from its equilibrium position. Indicates loudness/brightness/strength, but not speed. Wavelength ( ) Shortest distance between two points in the same phase (crest to crest, compression to compression). Frequency ( f ) and Period ( T ) Frequency f is the number of cycles per second; period T is the time for one cycle. They obey f=1/T . Angular Frequency ( ) Rate of phase change in radians per second: =2 f . Spatial frequency in radians per metre: k=2 / . Wavenumber ( k ) Phase The argument of the sinusoid (e.g., kx- t+ ) that sets the state of oscillation at a point: crest, trough, or in-between. A surface (in 3D) or line (in 2D) joining points with the same phase (e.g., all crests). Spherical wavefronts expand from a point source; plane wavefronts are parallel lines. Wavefront Phase Velocity ( v p ) The speed at which a point of fixed phase (e.g., a crest) travels: v p= /k=f . The most useful model for a one-dimensional progressive wave on a string or along a line of air particles is a sinusoid. A single formula tells you the displacement y of the medium at any position x and time t . Matching this to measurements lets you extract A , , f , and the direction of motion in seconds. Progressive wave (right-moving) A crest moves in the +x direction when the time term has a minus sign. tip Sign convention: kx- t travels in +x; kx+ t travels in −x. Think “Right is Minus” for the time term. To verify the direction, set the phase =kx- t+ to a constant (say crest at = /2 ). Increasing t then requires x to increase to keep fixed, so the crest moves toward +x. If the sign before t is positive, increasing t requires x to decrease, indicating motion toward −x. Frequency and wavelength are locked by the medium’s speed. Wave speed relations The wave speed is determined solely by the properties of the medium, regardless of the wave's frequency or amplitude. In a given medium under fixed conditions, v is set by the medium, not by how large A is. If you double f , the wavelength halves so that f stays equal to v . Most NEET questions hinge on manipulating v=f and identifying the correct v for the medium from a formula like T/ or P/ . Definitions Conversions between linear and angular measures. Units: k has units rad/m , has units rad/s , f is Hz , and is m . The amplitude A shares units with displacement (often metres). The phase constant is in radians and sets the starting state at t=0 and x=0 . Here T is tension and is linear mass density. Speed on a stretched string v= T Small slopes: | y/ x | 1 so y/ x Uniform tension T and linear density No stiffness (ideal flexible string), no damping Wave equation on a string and v= T/ neet-alert Use v= T/ only when the string is uniform, slopes are small, and tension is the same along its length. Adding heavy attachments or large amplitudes breaks the model. Dependence is intuitive: more tension T tightens the string, increasing v ; more mass per length makes it sluggish, decreasing v . For a family of experiments on the same string, v T . This square-root law is often tested by doubling T and asking how v and change for a given frequency. Square root of tension (√T) √N On a fixed string, plotting v against √T gives a straight line (slope = 1/√μ). Wave speed v m/s custom A straight line through the origin showing v ∝ √T for fixed μ. control dependent control is ratio of specific heats, P is pressure, density, R universal gas constant, T absolute temperature, M molar mass. Speed of sound in a gas This formula calculates the speed of sound in an ideal gas under adiabatic conditions, depending on the gas's properties and pressure. v= P = R T M Newton–Laplace result for sound speed in gases Small amplitude plane sound waves Adiabatic compressions/expansions in the gas Ideal gas behavior for the R T/M form neet-alert Always use absolute temperature: T in kelvin. At fixed gas and small pressure range, v T . If humidity is involved, effective molar mass decreases and v increases slightly. For air near room conditions, take 1.4 , M 29 , g/mol , so v 331 , m/s at 0 C and increases about 0.6 m/s per °C. In water and steel, sound travels much faster because the bulk modulus B is enormous compared to air, despite higher density. B is bulk modulus (stiffness to compression). In solids, a longitudinal modulus may be used; transverse (shear) waves use shear modulus. Speed in liquids/solids Determine the speed of a transverse wave using the ratio of applied tension to the string's linear mass density. As stiffness (bulk or shear modulus) rises, v increases; as density rises, v decreases. That is why sound speed in steel ( 5 10 3 , m/s ) is far larger than in air, even though steel is denser: the massive increase in stiffness dominates the effect of density. String (transverse) v = T/ ↑ with T, ↓ with μ 50–300 m/s (guitar, sonometer) Gas (sound) v = P/ = RT/M ↑ with T, ↓ with M ≈ 340 m/s (air, 20 C) Liquid (sound) v = B/ ↑ with B, ↓ with ρ ≈ 1500 m/s (water) Solid (longitudinal) v = B/ (or elastic modulus/ρ) ↑ with stiffness, ↓ with ρ ≈ 5000 m/s (steel) Speed formulas across media Medium/Wave Speed formula Key dependence Typical values Wave types often appear in classification questions. Mechanical waves (strings, sound) need a medium; electromagnetic (light, radio) do not. Transverse waves have oscillations perpendicular to propagation; longitudinal waves have oscillations parallel to propagation. Sound in air is longitudinal; waves on a string are transverse. Electromagnetic waves are transverse and can be polarised; longitudinal waves cannot be polarised. Transverse vs Longitudinal vs Electromagnetic Particle motion Perpendicular to propagation Parallel to propagation Perpendicular electric and magnetic fields Medium required Yes Yes No (propagate in vacuum) Polarisation Possible Not possible Possible Common examples String waves, water surface ripples Sound in air, compression waves in springs Light, radio, X-rays Property Transverse (Mechanical) Longitudinal (Mechanical) Electromagnetic A bigger amplitude makes a wave travel faster. In linear media, speed depends on medium properties (T, μ, B, ρ, γ), not amplitude. v=f ties f and once v is set. In y=A (kx- t) the negative sign means the displacement is negative. The sign sets direction of motion of the pattern, not the sign of y . The instantaneous y depends on the sine value at that phase. remember Waves transfer energy, not matter. Particles oscillate around equilibrium; only the phase (crests, compressions) travels. Wavefronts give a clean geometric picture. Imagine a point source of sound: at each instant, crests form a spherical shell expanding outward. Far away from the source, that shell looks almost flat in a small region: the wavefront is approximately plane. Rays drawn perpendicular to wavefronts show the local direction of propagation. A snapshot y(x,t 0) shows shape in space at an instant; a time trace y(x 0,t) shows oscillation at a point. For a sinusoid, the spatial graph is a sine curve with period ; the time graph is a sine curve with period T=1/f . Reading these correctly is a frequent exam skill. A sine curve snapshot at fixed time t0 illustrating one full wavelength between successive crests. control dependent t0 control Position x Displacement y Spatial profile y(x) at an instant. Distance between crests is λ. Node Crest λ/4 Zero crossing λ/2 −A 3λ/4 Trough custom Standing waves form when two identical waves travel in opposite directions and superpose. On a string with both ends fixed, allowed wavelengths satisfy n=2L/n and f n=nv/(2L) . The medium’s speed v still comes from T/ , so tension or mass density changes shift all harmonics together. Standing wave as the sum of two counter-propagating waves: y=2A (kx) ( t) . Boundary conditions pick out which standing wave patterns fit. Fixed ends enforce nodes at the ends; open ends of an air column allow antinodes. Regardless of pattern, the local wave speed continues to be set by the medium properties ( T, for strings; B, for sound). m/s, m String speed and wavelength relation A string has tension T=36 , N and linear density =4.0 , g/m . A wave of frequency f=120 , Hz travels along it. Find the wave speed and wavelength. easy Use v= T/ and =v/f . T = 36 N μ = 0.004 kg/m f = 120 Hz Wave speed v and Wavelength λ The speed depends only on T and . Changing the driving frequency would change so that f =v still holds. This is a classic check: if f doubles, the wavelength halves for the same string. Gas sound speed vs temperature Estimate the speed of sound in oxygen gas at 27 C (300 K). Take =1.4 , M=32 , g/mol . Speed of sound v Use v= R T/M . γ = 1.4 T = 300 K M = 0.032 kg/mol R = 8.314 J/(mol·K) medium m/s For the same temperature, a lighter gas (smaller M ) has a higher speed of sound. That is why sound travels slightly faster in moist air than in dry air: the effective molar mass is lower when water vapour (18 g/mol) replaces some heavier molecules. m, Hz, m/s Reading parameters from a wave equation A transverse wave on a string is given by y=0.010 (4 t-2 x) where x is in metres and t in seconds. Find (i) amplitude, (ii) , (iii) f , (iv) v and direction, and (v) particle velocity at x=0.25 , m , t=0.125 , s . A, λ, f, v and direction, v y at given x,t y = 0.010 cos(4π t − 2π x) m Compare to y=A ( t-kx) rewritten as A (4 t-2 x) . Use =4 , k=2 , v= /k . hard A common slip is to mix up the signs and conclude the wave goes left when it actually goes right. Always match your equation with kx t carefully, or track a fixed phase point as t increases. A string of length L=1.2 , m is fixed at both ends. Tension T=100 , N , linear density =0.010 , kg/m . Find the frequency of the 3rd harmonic and the number of nodes. Harmonics on a string with known speed f3 and number of nodes Compute v= T/ then f n=nv/(2L) . L = 1.2 m T = 100 N μ = 0.010 kg/m n = 3 medium Hz All harmonics scale together with v . Any change in tension or linear density shifts the entire harmonic series by the same factor. Length L sets the spacing between allowed modes. tip Model limits: Linear wave equations assume small amplitudes and linear restoring forces. Very large amplitudes can introduce dispersion and nonlinearity, making speed depend weakly on amplitude. Direction memory: “Right is Minus” (y = A sin(kx − ωt) goes +x). Triangle memory: put v on the top and f, λ at the bottom so v = fλ, f = v/λ, λ = v/f. Sound must be slower in solids because solids are denser. Speed depends on stiffness and density: v = √(B/ρ). Solids have huge stiffness (B), so v is usually much larger despite higher density. Reading data from wave graphs: If you have a spatial snapshot, measure the distance between two successive crests to get ; if you have a time trace at a point, measure the time between crests to get T and hence f=1/T . Combine with the correct medium speed formula to solve for unknowns. Always keep units consistent. Energy and power in waves: Although particles only oscillate locally, energy flows with the wave. For a string, average power is proportional to A 2 2 v . This shows why a larger amplitude can carry more energy even though the phase speed v is unchanged by amplitude in linear media. Phase difference matters: Two points separated by x on the same snapshot have =k x . Two instants separated by t at the same point have = t . These relations let you translate spatial and temporal measurements into the phase language used by wave equations. Units sanity checks for NEET: T in newtons, in kg/m, B and P in pascals, in kg/ m 3 , R=8.314 , J ,mol -1 ,K -1 , M in kg/mol, T in kelvin. A quick unit check often catches hidden mistakes like using Celsius or grams without conversion. Left-moving waves: If the equation is written as y(x,t)=A (kx+ t+ ) , the pattern moves toward negative x . You can always confirm by holding the phase fixed and noting which way x must change as t increases. Left-moving progressive wave Crests move in the −x direction for the plus sign. This equation describes the displacement pattern of a linear, non-dispersive wave traveling in the positive x-direction. Practical measurement tip: To find of a string, measure length and mass and compute =m/L . Then adjust tension with a known weight on a pulley (T ≈ mg for a light, frictionless pulley) to set the desired wave speed for resonance experiments. neet-alert Do not plug Celsius into v= RT/M . Convert to kelvin first. Also, use consistent molar mass units: kg/mol for SI. Sound vs electromagnetic waves: Light in vacuum always travels at c 3.0 10 8 , m/s , independent of frequency. In media, light can be dispersive (speed depends on frequency), unlike the ideal string or ideal gas approximations used here. NEET most often treats mechanical waves as nondispersive unless stated. Observed frequency shifts when source or observer moves in a medium with speed v: f obs =f s v+v o v-v s (sound). Although Doppler effect concerns observed frequency rather than the intrinsic wave speed, it presumes you know the medium’s v to compute and f obs . The medium sets v ; motion shifts the relative spacing of wavefronts reaching the observer. Wave pulses vs periodic waves: A single pulse also travels at the medium’s speed determined by T and (string) or B and (sound). The periodic sine wave is just a convenient steady pattern for analysis; the speed law comes from the medium, not the repeating nature. Edge cases: At extremely low pressures or in rarefied gases, the continuum assumptions behind v= P/ can fail. In very stiff strings (piano wires), dispersion due to stiffness slightly increases higher-mode frequencies beyond the ideal n pattern; basic NEET items ignore this unless explicitly mentioned. Amplitude Maximum displacement of the medium, A . Spatial period of the wave, . Wavelength Cycles per second, f , with f=1/T . Frequency Angular frequency Phase rate in rad/s, =2 f . Spatial angular frequency, k=2 / . Wavenumber v p Phase velocity Speed of a fixed phase point, v= /k=f . Mass per unit length of a string, =m/L . Linear density Bulk modulus Stiffness to compression, B=-( P)/( V/V) . Wavefront Surface/line connecting equal-phase points (e.g., all crests). Oscillations perpendicular to propagation. Transverse wave Longitudinal wave Oscillations parallel to propagation. Standing wave Stationary pattern from two opposite waves: y=2A (kx) ( t) . Quick glossary