Introduction to waves#


In this notebook, wave properties will be discussed, as well as derivations for simple harmonic motion and the wave equation.

import numpy as np
import matplotlib.pyplot as plt

Wave properties#

A summary of the wave properties can be seen in the figure below:

image info

Selection of wave properties:

  • \(t\) – time

  • \(A\) – amplitude

  • \(f\) – frequency, \(f = \frac{\omega}{2\pi}\)

  • \(\lambda\) – wavelength, \(\lambda = vf = \frac{2\pi}{k}\)

  • \(T\) – period, \( T = \frac{1}{f}\)

  • \(p\) – phase shift

  • \(\omega\) – angular frequency

  • \(k\) – wave number

  • \(c_p\) – phase speed, \(c_p = \frac{\omega}{k}\)

A variation and comparison between a selection of wave properties can be seen below:

def sine_wave(t, A, f, p):
    This formula will be derived later on for simple harmonic motion using Newton's 2nd law
    y = A*np.sin(2*np.pi*f*t + p*np.pi/180)
    return y

t = np.linspace(0,5,200)

A1 = 2
f1 = 1
p1 = 1
y1 = sine_wave(t, A1, f1, p1)

A2 = 1.5
f2 = 2
p2 = 0.2
y2 = sine_wave(t, A2, f2, p2)

plt.xlabel("Time (s)")
plt.plot(t, y1)
plt.plot(t, y2)

Simple Harmonic Motion#

A wave carries information and energy but does not transport the material. It undergoes simple harmonic motion.

To derive the formula for simple harmonic motion (SHM), assume an object of mass m and mean position \(y_0\) and the displacement ‘\(y\)’. In SHM, there is a restoring force that is directed towards the mean position \(y_0\) at every instant. The restoring force is directly proportional to the displacement of the body at every instant, so \(F \propto -y\). The negative sign indicates that the restoring force acts opposite to the displacement. Thus, including the force constant, \(k\),

\[F = -ky \qquad (1)\]

Now consider Newton’s second law of motion \(F = ma\), where \(F\) is the restoring force on the body, \(m\) is the mass and \(a\) is the acceleration.

Since \(a = \frac{dv}{dt}= \frac{d^2 y}{d t^2}\), so

\[F = m \frac{d^2 y}{d t^2} \qquad (2)\]

Combining equations (1) and (2), we get

\[m \frac{d^2 y}{d t^2} = -ky\]
\[\frac{d^2 y}{d t^2} = - \frac{k}{m}y\]
\[\frac{d^2 y}{d t^2} + \frac{k}{m}y = 0\]

Let \(\frac{k}{m} = \omega^2\). Now,

\[\frac{d^2 y}{d t^2} + \omega^2 y = 0 \qquad (3)\]

The solution to this equation can be approximated from analysis, knowing that a sin or cosine function when differentiated twice returns to it’s original form, with a factor in front. So, we can estimate the solution to be

\[ y = A \sin(\omega t)\]

As we know that

\[v = \frac{dy}{dt} = A\omega \sin(\omega t)\]


\[a = \frac{d^2y}{dt^2} = A\omega^2 \sin(\omega t)\]

Filling \( y = A \sin(\omega t)\) back into equation (3), we get

\[ A\omega^2 \sin(\omega t) + A \sin(\omega t) \cdot \omega^2\]

which shows that this solution is valid.

Thus, the solution to the wave equation is:

\[ y = A \sin(\omega t) = A \sin (2 \pi f t)\]

Here, \(A\) is the magnitude of the initial displacement, \(\omega\) is the angular frequency and \(t\) is time.

def displacement(t, A, f):
    y = A*np.sin(2*np.pi*f*t)
    return y

def velocity(t, A, f):
    v = A*2*np.pi*f*np.cos(2*np.pi*f*t)
    return v

def acceleration(t, A, f):
    a = A*(2*np.pi*f)**2*np.sin(2*np.pi*f*t)
    return a

t = np.linspace(0,5,200)

A = 2
f = 1
displacement = displacement(t, A, f)
velocity = velocity(t, A, f)
acceleration = acceleration(t, A, f)

plt.xlabel("Time (s)")
plt.plot(t, displacement, label = "displacement (m)")
plt.plot(t, velocity, label = "velocity (m/s)")
plt.plot(t, acceleration, label = "acceleration ($m^2$/s)")
plt.title("Relationship of displacement, velocity and acceleration")

Note a few things in the plot above:

  • Velocity is zero where particle displacement is at it’s maximum or minimum.

  • Velocity is at it’s maximum or minimum when the particle displacement goes through the equilibrium position.

  • When particle displacement is at the equilibrium position, acceleration is zero.

  • Acceleration has a maximum or minimum when the particle displacement has a maximum or minimum.

Using trigonometry,

\[F = ma = -mg\sin(\theta)\]


\[a = g\sin(\theta)\]

Approximate that for small \(\sin(\theta) = \theta\), so

\[F = ma = mg\theta = m \frac{y}{l}\]


\[ \begin{align}\begin{aligned}a = \frac{F}{ml}y$$ (tussenstap). \\If the restoring force is tension from a string for example, it can be approximated as $F=-2T$, considering a string pulls the body from the left and right side.\\The equation of simple harmonic motion then becomes: \\$$\frac{d^2 y}{d t^2} = - \frac{2T}{ml} y = -\omega ^2 y \end{aligned}\end{align} \]

Wave equation#

To derive the wave equation, a string with density \(\rho\) per unit length is stretched in the \(x\) direction with tension force \(T\). A wave can be described as coupled simple harmonic motion. This can be thought of as a rope with lots of masses being displaced individually.

wave equation

The vertical forces on the string must balance.

Consider Newton’s second law for the force, where \(F=ma\)

Consider \(m=\rho\Delta x\), as \(m = \rho V\) and only one dimension is considered. So, force is

\[ma=\rho \Delta x \cdot \frac{\partial ^2 u}{\partial t^2}\]

This is equal to the difference between the vertical tension T at the two ends, so

\[\rho \Delta x \cdot \frac{\partial ^2 u}{\partial t^2} = T(x+ \Delta x, t)\sin\left[\theta(x+\Delta x, t)\right] - T(x,t)\sin\left[\theta(x,t)\right]\]

Assume that the deflection angles \(\theta\) are small, \(\sin\theta = \tan\theta = \frac{\partial u}{\partial x}\). Thus,

\[\rho \Delta x \cdot \frac{\partial ^2 u}{\partial t^2} = T(x+ \Delta x, t)\frac{\partial u}{\partial x} - T(x,t)\frac{\partial u}{\partial x}\]

Divide both sides by \(\Delta x\) and letting \(\Delta x \rightarrow 0\) gives:

\[\rho(x)\frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial x}\left(T \frac{\partial u}{\partial x}\right)\]

Assuming deflection is small, \(T\) can be assumed to be constant.

Considering \(c^2 = \frac{T}{\rho}\), the wave equation becomes

\[\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\]

One solution to the wave equation is by using travelling waves of the form \(u(x,t) = f(x \pm ct)\).

At \(t=0\), \(f(x)\) is a shape in space.

\(f(x-ct)\) is the shape of \(f(x)\) when travelling to the right at speed \(c\).

\(f(x+ct)\) is the shape of \(f(x)\) when travelling to the left at speed \(c\).

wave equation

To plot the wave equation, a few initial conditions have to be set. consider

Types of waves#

Body waves

  • Longitudinal waves: waves where the displacement of the medium is in the same direction as the travelling wave. Examples are P-waves or sound waves.

  • Transverse wave: waves where motion of all points oscillate in a path that is perpendicular to the direction of the travelling wave. Examples are seismic S-waves, electro-magnetic waves and vibrations on string.

wave equation

Surface waves

  • Love wave: wave moving horizontally and perpendicular to the direction of travel.

wave equation

  • Rayleigh wave: wave causing elliptical retrograde motion (opposite to direction of travel) in the vertical plane along direction of travel as a combination of both longitudinal and transverse vibration.

wave equation

  • Water wave: circular motion of elements moving in the same direction of wave travel.

wave equation


Imperial College waves lecture notes