understand and use the terms displacement, amplitude, phase difference, period, frequency, wavelength and speed

Progressive Waves

In this topic we will define the fundamental quantities that describe a progressive (traveling) wave and learn how they are related. By the end of the lesson you should be able to use the terms displacement, amplitude, phase difference, period, frequency, wavelength and speed correctly in equations and explanations.

Key Terms at a Glance

1. Displacement and Amplitude

The displacement \$y(x,t)\$ of a particle in a medium varies with both position \$x\$ and time \$t\$. For a simple sinusoidal wave travelling in the +\$x\$ direction the displacement can be written as

\$y(x,t) = A \sin\!\bigl(kx - \omega t + \phi_0\bigr)\$

where \$A\$ is the amplitude, \$k = \dfrac{2\pi}{\lambda}\$ is the wave‑number and \$\omega = 2\pi f\$ is the angular frequency. The amplitude is the maximum absolute value of \$y\$.

2. Phase Difference

The argument of the sine function, \$kx - \omega t + \phi_0\$, is called the phase. Two points are said to be in phase when their phase difference \$\Delta\phi\$ is an integer multiple of \$2\pi\$.

• If \$\Delta\phi = 0\$ the points are exactly in phase (e.g., two crests).
• If \$\Delta\phi = \pi\$ the points are opposite in phase (crest opposite trough).
• General relation: \$\Delta\phi = k\,\Delta x - \omega\,\Delta t\$.

3. Period and Frequency

The period \$T\$ is the time taken for a given point on the wave to complete one full oscillation. Frequency \$f\$ is the reciprocal of the period:

\$f = \frac{1}{T}\$

Both quantities describe the temporal behaviour of the wave and are measured at a fixed spatial location.

4. Wavelength

The wavelength \$\lambda\$ is the spatial analogue of the period. It is the distance over which the wave repeats its shape. For a sinusoidal wave the distance between two successive crests (or any two points in phase) is \$\lambda\$.

5. Wave Speed

The speed \$v\$ of a progressive wave links the temporal and spatial characteristics:

\$v = f\lambda = \frac{\lambda}{T} = \frac{\omega}{k}\$

Thus, if any two of the quantities \$f\$, \$\lambda\$, \$v\$ are known, the third can be found directly.

6. Example Problem

1. Given a wave with \$f = 50\ \text{Hz}\$ and \$\lambda = 0.2\ \text{m}\$, calculate its speed.
2. Use \$v = f\lambda\$.
3. \$v = 50\ \text{Hz} \times 0.2\ \text{m} = 10\ \text{m s}^{-1}\$

7. Summary Checklist

• Identify the displacement function \$y(x,t)\$ for a given wave.
• Read off the amplitude \$A\$ from the maximum displacement.
• Determine phase differences using \$\Delta\phi = k\Delta x - \omega\Delta t\$.
• Convert between period \$T\$ and frequency \$f\$.
• Measure or calculate wavelength \$\lambda\$ from the spatial pattern.
• Apply \$v = f\lambda\$ to find the wave speed.