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Mathematical Wave Descriptions

Level 1 - Physics topic page in Mechanical Waves.

Principle

A traveling wave is a fixed shape shifted through space as time passes.

The important step is to turn that geometric shift into an equation before specializing to sinusoids.

Notation

\(y(x,t)\)
wave displacement
\(\mathrm{m}\)
\(A\)
amplitude
\(\mathrm{m}\)
\(k\)
wavenumber
\(\mathrm{rad\,m^{-1}}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(\phi\)
phase constant
\(\mathrm{rad}\)
\(v_w\)
wave speed
\(\mathrm{m\,s^{-1}}\)

Method

Derivation 1: Shift a profile to the right or left

Start with a shape \(g(x)\) at \(t=0\). If that exact shape moves right with speed \(v_w\), every feature originally at \(x\) appears later at \(x+v_wt\).

Right-moving profile
\[y(x,t)=g(x-v_wt)\]
Left-moving profile
\[y(x,t)=g(x+v_wt)\]
Reason
\[x-v_wt=C\]
A constant phase point travels with the wave.

The two curves below have the same shape. The later curve is the original profile shifted to the right, which is exactly what the argument \(x-v_wt\) means.

02468-1.2-0.600.61.2xyt=0later

Derivation 2: Choose a sinusoidal profile

For a sinusoidal wave, the phase must change by \(2\\pi\) over one wavelength and over one period.

Spatial phase rate
\[k=\frac{2\pi}{\lambda}\]
Temporal phase rate
\[\omega=\frac{2\pi}{T}\]
Right-moving sinusoid
\[y(x,t)=A\cos(kx-\omega t+\phi)\]
Left-moving sinusoid
\[y(x,t)=A\cos(kx+\omega t+\phi)\]

Derivation 3: Recover the wave speed from the phase

A fixed phase point satisfies \(kx-\\omega t+\\phi=C\). Differentiating that condition gives the propagation speed.

Constant phase
\[k\,dx-\omega\,dt=0\]
Phase speed
\[\frac{dx}{dt}=\frac{\omega}{k}\]
Wave-speed form
\[v_w=\frac{\omega}{k}\]

Rules

These are the compact results from the method above.

Right-moving wave
\[y(x,t)=g(x-v_wt)\]
Left-moving wave
\[y(x,t)=g(x+v_wt)\]
Sinusoidal wave
\[y(x,t)=A\cos(kx-\omega t+\phi)\]
Wavenumber
\[k=\frac{2\pi}{\lambda}\]
Angular frequency
\[\omega=\frac{2\pi}{T}\]
Phase speed
\[v_w=\frac{\omega}{k}\]

Examples

Question
For
\[y(x,t)=0.040\cos(5x-20t)\]
find the amplitude, wavenumber, angular frequency, speed, and direction of travel.
Answer
The amplitude is
\[0.040\,\mathrm{m}\]
the wavenumber is
\[k=5\,\mathrm{rad\,m^{-1}}\]
and the angular frequency is
\[\omega=20\,\mathrm{rad\,s^{-1}}\]
The speed is
\[v_w=\frac{\omega}{k}=\frac{20}{5}=4.0\,\mathrm{m\,s^{-1}}\]
The form
\[kx-\omega t\]
shows a right-moving wave.

Checks

  • The sign in the phase determines direction: \(x-v_wt\) moves right and \(x+v_wt\) moves left.
  • \(k\) measures phase change per meter, not meters per cycle.
  • \(v_w=\\omega/k\) is the same speed as \(f\\lambda\).
  • A sinusoid is one special traveling profile, not the definition of all waves.

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Wave Speed