Mechanical Waves and Sound
The wave equation, \(\dfrac{\partial^2y}{\partial t^2}=v^2\dfrac{\partial^2y}{\partial x^2}\), is this course's one genuine step into partial derivatives — a function of two variables, position and time, differentiated with respect to each separately. Verifying that \(y(x,t)=A\sin(kx-\omega t)\) satisfies it is the exact same substitution-and-check technique used for the SHM and drag equations, just with one more variable in play.
What you'll learn
- Take partial derivatives of a two-variable wave function y(x,t) with respect to x and t separately
- Verify that y(x,t)=A sin(kx−ωt) satisfies the wave equation, and identify wave speed as v=ω/k
- Connect wavelength, frequency, and wave speed (v=fλ) to the k and ω in a wave function
1. Partial Derivatives of a Wave Function
A traveling wave, \(y(x,t)\), depends on both position and time — differentiating it means choosing which variable to differentiate with respect to, holding the other fixed. \(\partial y/\partial t\) (holding \(x\) fixed) gives the transverse velocity of one specific point on the medium; \(\partial y/\partial x\) (holding \(t\) fixed) describes the wave's shape at one frozen instant. Both are needed for the wave equation, and neither is the same as the wave's own speed moving through the medium.
2. The Wave Equation
A function \(y(x,t)\) describes a wave traveling at speed \(v\) exactly when it satisfies \(\dfrac{\partial^2y}{\partial t^2}=v^2\dfrac{\partial^2y}{\partial x^2}\). Verifying that a specific \(y(x,t)=A\sin(kx-\omega t)\) solves this is the same substitution-and-check technique from Units 102 and 106, just requiring both partial derivatives instead of one ordinary derivative.
3. Wave Speed, Wavelength, and Frequency
The parameters \(k\) and \(\omega\) in a wave function connect directly to the more familiar wavelength and frequency: \(\lambda=2\pi/k\), \(f=\omega/2\pi\), and \(v=f\lambda=\omega/k\) — the same relationship derived above, just relabeled in terms students may already recognize from an algebra-based treatment of waves.
Key equations
- ∂²y/∂t² = v²·∂²y/∂x² — The wave equation — any y(x,t) satisfying this describes a wave traveling at speed v.
- y(x,t) = A sin(kx − ωt) — A traveling sinusoidal wave — verified by substitution to solve the wave equation, with wave speed v=ω/k.
- v = ω/k = fλ — Wave speed in terms of the wave function's parameters, or equivalently in terms of frequency and wavelength.