Difficulty: Introductory to Intermediate | Prerequisites: Section 15.1 (wave model, transverse vs longitudinal). Basic trigonometry (sine, cosine) is needed for the displacement equations.
Once you know what a wave is (15.1), the next question is how fast it moves and how to describe its shape mathematically. Sections 15.2 through 15.4 give you the quantitative toolkit: wave speed formulas, the sinusoidal displacement equation, and two types of graph (snapshot and history) that let you read off amplitude, wavelength, period and frequency at a glance. These tools recur constantly in later chapters on sound, standing waves and interference. If you can write and interpret the displacement equation y(x, t), the rest of wave mechanics becomes algebra.
Wave speed depends only on the medium, not on amplitude or frequency. You can capture a wave's shape at one instant (snapshot graph) or track one point over time (history graph), and both give you the same displacement equation y(x, t) = A sin(kx - wt) once you read off the right parameters.
Linear density (mu)
The mass per unit length of a string or rope: mu = M / L. It tells you how "heavy" each metre of the string is. A thicker or denser string has a higher linear density, and waves travel more slowly on it.
Wave speed (v)
The speed at which the wave disturbance propagates through the medium. It depends only on the properties of the medium, not on amplitude or frequency.
Wavelength (lambda)
The distance between two successive identical points on the wave (e.g. crest to crest). In simple terms, it is the spatial length of one full cycle.
Amplitude (A)
The maximum displacement of a particle from its equilibrium position. It determines how "tall" the wave is and relates to the wave's energy.
Frequency (f)
The number of complete oscillations per second, measured in hertz (Hz). Think of it as how many crests pass a fixed point each second.
Period (T)
The time for one complete oscillation: T = 1/f. It is the temporal length of one cycle.
Wave number (k)
k = 2 pi / lambda. It counts how many radians of phase fit into one metre of space. Think of it as the spatial equivalent of angular frequency.
Angular frequency (omega)
omega = 2 pi f = 2 pi / T. It gives the rate of oscillation in radians per second rather than cycles per second.
Snapshot graph
A graph of displacement y versus position x at a single frozen moment in time. From it you read amplitude and wavelength.
History graph
A graph of displacement y versus time t at a single fixed position. It is a reflection (in shape) of the snapshot graph. From it you read amplitude and period.
Crest
The point of maximum positive displacement on a wave.
Trough
The point of maximum negative displacement on a wave.
Increasing the tension in a string increases wave speed.
Increasing the linear density (more mass per unit length, i.e. more inertia) decreases wave speed.
For sound: waves move faster in liquids than in gases, and faster still in solids. This is because denser, stiffer media restore displaced particles more quickly.
The speed of sound in air at 20 degrees C is approximately 343 m/s.
Snapshot graph (y vs x at fixed t): freezes the wave at one moment. The horizontal axis is position. You read wavelength (lambda) from crest-to-crest distance and amplitude (A) from the peak displacement.
History graph (y vs t at fixed x): watches one point on the medium over time. The horizontal axis is time. You read period (T) from peak-to-peak time interval and amplitude from the peak displacement.
The history graph looks like a reflection of the snapshot graph. A common exam task is to derive one from the other.
Key distinction: the dot (a specific point on the string) is not moving along with the wave. The wave moves through the string's position; the dot simply oscillates up and down (or back and forth for longitudinal).
Wave moving to the right (positive x-direction):
y(x, t) = A cos(2 pi (x/lambda - t/T))
Equivalently: y(x, t) = A sin(kx - omega t)
Wave moving to the left (negative x-direction):
y(x, t) = A cos(2 pi (x/lambda + t/T))
Equivalently: y(x, t) = A sin(kx + omega t)
The sign between the spatial and temporal terms tells you the direction: minus for positive direction, plus for negative direction.
v = lambda f = lambda / T
k = 2 pi / lambda
omega = 2 pi f = 2 pi / T
omega = k v (this links angular frequency, wave number and wave speed)
f = omega / (2 pi)
Wave speed on a string:
v_s = sqrt(T_s / mu)
where T_s is the tension in the string and mu is the linear density (mass/length).
Speed of sound in an ideal gas (molecular form):
v_rms = sqrt(3 k_B T / m)
where k_B is Boltzmann's constant, T is absolute temperature in kelvin, and m is the molecular mass.
Speed of sound in an ideal gas (molar form):
v = sqrt(gamma k_B T / m) = sqrt(gamma R T / M)
where gamma = 1.67 for monatomic gases, 1.4 for diatomic, approximately 1.3 for triatomic. R is the gas constant and M is the molar mass (kg/mol).
Fundamental wave relationships:
v = lambda f = lambda / T
k = 2 pi / lambda
omega = 2 pi f = 2 pi / T = k v
Displacement (rightward wave):
y(x, t) = A sin(kx - omega t)
Displacement (leftward wave):
y(x, t) = A sin(kx + omega t)
Musical instrument strings illustrate wave speed on a string directly: a thicker bass guitar string (higher linear density) produces a lower-frequency note at the same tension, precisely because waves travel more slowly along it. Tuning pegs adjust tension to change wave speed, which changes pitch.
The temperature dependence of the speed of sound matters in aviation and meteorology. Pilots correct altimeter readings for non-standard temperatures because the speed of sound, and therefore the behaviour of pressure waves around the aircraft, changes with air temperature.
Students often think that increasing the amplitude or frequency of a wave changes its speed. It does not. Wave speed is set by the medium alone.
Confusing a snapshot graph with a history graph is one of the most common exam errors. Check the horizontal axis: if it is x (position), it is a snapshot; if it is t (time), it is a history graph.
Students sometimes believe a point on the string moves along with the wave. The point oscillates in place; the wave pattern moves through it.
Mixing up the sign convention in the displacement equation is another frequent mistake. For a wave moving in the positive x-direction, the argument is (kx - omega t). A plus sign means the wave moves in the negative x-direction.
Expect to be given a snapshot or history graph and asked to extract A, lambda, T, f, v or write the displacement equation. Practise this until it is automatic.
The formula v = sqrt(T_s / mu) is a staple calculation question. Know what happens when you double the tension (speed goes up by a factor of sqrt(2)) or double the linear density (speed goes down by a factor of sqrt(2)).
The sign convention in y(x, t) = A sin(kx +/- omega t) is a reliable exam differentiator. Many students lose marks here.
Be ready to convert between f, T, lambda, k, omega and v fluently. A question may give you two of these and ask for the rest.
True or false: Doubling the frequency of a wave on a string doubles the wave speed.
False. Wave speed depends on the medium (tension and linear density), not frequency.
Fill in the blank: On a snapshot graph, the horizontal axis is ______, and you can read off the ______.
Position (x); wavelength (lambda).
True or false: A minus sign in the argument (kx - omega t) indicates a wave moving in the positive x-direction.
True.
Fill in the blank: omega = 2 pi / ______.
T (the period).
True or false: Sound travels faster in air than in water.
False. Sound travels faster in water (and faster still in solids).
Q: A string has a mass of 0.040 kg and a length of 2.0 m. If the tension is 50 N, what is the wave speed?
A: mu = 0.040 / 2.0 = 0.020 kg/m. v = sqrt(50 / 0.020) = sqrt(2500) = 50 m/s.
Q: A wave on a string has a wavelength of 0.80 m and a frequency of 25 Hz. What is the wave speed?
A: v = lambda f = 0.80 x 25 = 20 m/s.
Q: Write the displacement equation for a wave with amplitude 0.10 m, wavelength 2.0 m and frequency 5.0 Hz, moving in the positive x-direction.
A: k = 2 pi / 2.0 = pi rad/m. omega = 2 pi (5.0) = 10 pi rad/s. y(x, t) = 0.10 sin(pi x - 10 pi t) metres.
Q: You are given a history graph that shows a period of 0.25 s and an amplitude of 3 cm. The wave is moving at 8 m/s. What is the wavelength?
A: f = 1 / T = 1 / 0.25 = 4 Hz. lambda = v / f = 8 / 4 = 2.0 m.
Q: In the equation y(x, t) = 0.05 sin(4x + 20t), in which direction is the wave moving and what is the wave speed?
A: The plus sign between the spatial and temporal terms means the wave moves in the negative x-direction. v = omega / k = 20 / 4 = 5 m/s.
The displacement equation y(x, t) is the foundation for superposition, interference and standing waves in later chapters. Two waves described by these equations, added together, give constructive or destructive interference depending on their relative phase.
The wave speed formula v = sqrt(T_s / mu) reappears when you study harmonics on a string: the allowed frequencies of a vibrating guitar string are directly determined by this speed and the string's length.
wave speed, linear density, mu, tension, wavelength, lambda, amplitude, frequency, period, wave number, angular frequency, omega, snapshot graph, history graph, displacement equation, sinusoidal wave, crest, trough, y(x t), phase, speed of sound, Boltzmann constant, gas constant, molar mass, gamma, travelling wave equation, General Physics Chapter 15, University of Washington physics