Source: PHYS 116, University of Washington, Sections 15.3–15.4
Difficulty: Introductory | Prerequisites: Basic algebra, familiarity with simple harmonic motion (SHM), understanding of frequency and period.
This material covers two foundational ideas in wave physics: how to describe waves mathematically (using graphs, equations and key parameters like wavelength and frequency), and how sound and light fit into the broader picture of wave phenomena. These sections sit early in the waves unit and set up everything that follows, including intensity, the decibel scale and the Doppler effect. You should already be comfortable with simple harmonic motion, since sinusoidal waves are built directly on that concept. If you can write down the displacement of an oscillating mass on a spring, you have the tools you need here.
Waves are described by snapshot graphs (displacement vs. position at one instant) and history graphs (displacement vs. time at one point). Sinusoidal waves use cosine/sine functions with wavelength, period, amplitude and wave number as parameters. Sound waves are longitudinal pressure waves; light and other electromagnetic waves are transverse and all travel at 3.00 × 10⁸ m/s in air.
Snapshot graph
A plot of the wave's displacement as a function of position at a single instant in time. Think of it as a photograph of the wave frozen at one moment.
History graph
A plot of the displacement of one particular point in the medium as a function of time. In simple terms, this means you pick one spot and watch how it bobs up and down as the wave passes.
Sinusoidal wave
A wave produced by a source that oscillates with simple harmonic motion (SHM). The displacement follows a cosine or sine function. Think of it as the smoothest, most regular wave shape you can get.
Wavelength (λ)
The distance between two successive crests (or troughs) of a wave, measured in metres. In simple terms, this is the length of one full wave cycle in space.
Period (T)
The time it takes for one complete wave cycle to pass a given point. Think of it as the time between two consecutive crests arriving at the same spot.
Amplitude (A)
The maximum displacement of the wave from its equilibrium position. In simple terms, this is how tall the wave is from the centre line to the peak.
Wave number (k)
Defined as k = 2π / λ. It tells you how many radians of phase fit into one metre of space. Think of it as the spatial equivalent of angular frequency.
Angular frequency (ω)
Defined as ω = 2πf, where f is frequency. It tells you how many radians of phase the wave cycles through per second.
Compression
A region of higher pressure in a longitudinal (sound) wave. Think of it as the crest of a sound wave.
Rarefaction
A region of lower pressure in a longitudinal (sound) wave. Think of it as the trough of a sound wave.
Electromagnetic wave
A transverse wave consisting of oscillating electric and magnetic fields. All electromagnetic waves travel through air (or vacuum) at c = 3.00 × 10⁸ m/s. Think of it as light and its relatives: radio waves, microwaves, infrared, ultraviolet, X-rays.
Wave front
A line (in 2D) or surface (in 3D) that connects all points of the wave at the same phase (typically the crests). In simple terms, this is the leading edge of a wave as it spreads outward.
Plane wave
A wave whose wave fronts are flat planes rather than curved surfaces. This is the approximation you use when you are very far from the source.
A wave is not localised at a single point. It exists throughout the entire medium through which it travels.
A wave pulse moves forward a distance Δx = vΔt during a time interval Δt, without changing shape.
Shows displacement as a function of position at one fixed instant.
It captures the shape of the wave along the medium at that moment.
A snapshot graph shows the motion of the wave, not the motion of the medium.
Shows displacement as a function of time at one fixed position.
It records how a single point in the medium moves as the wave passes through it.
Each point in the medium oscillates with SHM as a sinusoidal wave passes.
Displacement as a function of position at one instant:
y(x) = A cos(2π x/λ)
Displacement as a function of time at one point:
y(t) = A cos(2π t/T)
Displacement of a travelling wave moving to the right:
y(x, t) = A cos(2π(x/λ − t/T))
Displacement of a travelling wave moving to the left:
y(x, t) = A cos(2π(x/λ + t/T))
The displacement y is a function of two variables (x and t).
During exactly one period, each crest of a sinusoidal wave travels forward exactly one wavelength.
The fundamental relationship: v = λf
The wave number k = 2π/λ and angular frequency ω = 2πf give the alternate form:
y(x, t) = A sin(kx − ωt) for a wave moving to the right
Changing the amplitude does not change the wavelength or wave speed. These are independent properties.
Sound is a longitudinal pressure wave. The medium's particles oscillate parallel to the direction of wave travel.
Compressions are regions of higher pressure (analogous to crests). Rarefactions are regions of lower pressure (analogous to troughs).
A loudspeaker cone moves back and forth, creating alternating compressions and rarefactions in the air.
Shorter wavelength corresponds to higher frequency.
Human hearing range: approximately 20 to 20,000 Hz.
Other animals have different ranges. Dogs hear up to about 45,000 Hz; bats up to about 100,000 Hz; elephants can detect frequencies below 5 Hz.
All electromagnetic (EM) waves travel through air (and vacuum) at the same speed: c = 3.00 × 10⁸ m/s.
All EM waves are transverse waves, with oscillating electric and magnetic fields perpendicular to the direction of travel.
The electromagnetic spectrum spans from radio waves (long wavelength, low frequency) through microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays (short wavelength, high frequency).
Visible light occupies a narrow band roughly from 400 nm (violet) to 700 nm (red).
y(x) = A \cos\!\left(\frac{2\pi x}{\lambda}\right)Displacement as a function of position at one instant.
y(t) = A \cos\!\left(\frac{2\pi t}{T}\right)Displacement as a function of time at one point.
y(x,t) = A \cos\!\left(2\pi\!\left(\frac{x}{\lambda} - \frac{t}{T}\right)\right)Travelling wave moving to the right.
y(x,t) = A \cos\!\left(2\pi\!\left(\frac{x}{\lambda} + \frac{t}{T}\right)\right)Travelling wave moving to the left.
v = \lambda fWave speed equals wavelength times frequency.
k = \frac{2\pi}{\lambda}Wave number definition.
y(x,t) = A \sin(kx - \omega t)Alternate form using wave number and angular frequency (wave moving right).
\Delta x = v \Delta tDistance a wave pulse travels in time Δt.
Sound waves as pressure waves are the basis for how speakers, microphones and musical instruments work: a speaker cone creates compressions and rarefactions in air, and a microphone converts them back into electrical signals. The electromagnetic spectrum underpins everything from radio broadcasting and mobile phone signals (radio/microwave frequencies) to medical imaging (X-rays) and fibre-optic internet (infrared light pulses).
Students often confuse the snapshot graph with the history graph. The snapshot shows the whole wave at one moment; the history shows one point over time. They look similar but represent entirely different things.
Students sometimes think that increasing the amplitude will change the speed or wavelength of the wave. It does not. Amplitude, speed and wavelength are independent.
A common mistake is forgetting the sign convention in the travelling-wave equation: minus (x/λ − t/T) means the wave moves to the right, plus (x/λ + t/T) means it moves to the left.
Students sometimes describe sound as a transverse wave. Sound in air is longitudinal: the air molecules oscillate back and forth along the direction the wave travels, not perpendicular to it.
⚠️ Know how to read and distinguish snapshot graphs from history graphs. Exam questions often give you one type and ask you to sketch or interpret the other.
⚠️ Be able to write the travelling-wave equation for a wave moving in either direction, given A, λ and T. Pay close attention to the sign inside the cosine.
⚠️ Remember v = λf. You will use this relationship repeatedly in problems involving wave speed, frequency and wavelength.
⚠️ Know the key features of the EM spectrum, including the speed of all EM waves (3.00 × 10⁸ m/s) and the approximate range of visible light (400 to 700 nm). The speed of sound is different and medium-dependent.
True or false: a snapshot graph shows how a single point in the medium moves over time.
Fill in the blank: the wave number k is defined as k = ____.
True or false: all electromagnetic waves travel at the same speed in air.
True or false: doubling the amplitude of a sinusoidal wave doubles its speed.
Fill in the blank: in the travelling-wave equation, a minus sign between the x/λ and t/T terms means the wave is moving to the ____.
Q: A sinusoidal wave has a wavelength of 2.0 m and a frequency of 5.0 Hz. What is its speed?
A: v = λf = (2.0 m)(5.0 Hz) = 10 m/s.
Q: A wave on a string has amplitude 0.30 m, wavelength 4.0 m and period 0.50 s. Write the equation for this wave if it moves to the right.
A: y(x, t) = (0.30 m) cos(2π(x/4.0 − t/0.50)).
Q: A boat moves to the right at 5.0 m/s. An ocean wave moves to the left with A = 1.0 m, λ = 110 m and T = 8.3 s. At t = 0 the boat sits on a crest (x = 0). What is the boat's vertical displacement at t = 20 s?
A: At t = 20 s, the boat is at x = (5.0 m/s)(20 s) = 100 m. Using the leftward-wave equation: y = (1.0 m) cos(2π(100/110 + 20/8.3)) = −0.42 m.
Q: What is the wave number of a wave with wavelength 0.50 m?
A: k = 2π/λ = 2π/0.50 = 4π ≈ 12.6 rad/m.
Q: Is sound a transverse or longitudinal wave? What about light?
A: Sound (in air) is a longitudinal wave. Light is a transverse electromagnetic wave.
The sinusoidal wave equation builds directly on the simple harmonic motion you studied earlier: the cosine function and the parameters A, T and f carry over unchanged. This section's wave-description tools (snapshot and history graphs, the travelling-wave equation) are used throughout the rest of the waves unit, particularly when you study intensity (Section 15.5), the decibel scale (Section 15.6) and the Doppler effect (Section 15.7). The electromagnetic spectrum also connects to optics and modern physics later in the course.
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