Progressive Wave

Wave Pulse and Progressive Wave

• Drop a stone into still water: the water is displaced locally, the disturbance spreads, and distant particles are set oscillating about their positions of rest for a short time. They are not bodily carried along. Such a short-lived local disturbance is a wave pulse.
• A wave in which the disturbance produced in the medium travels in a given direction continuously, without any damping and obstruction, from one particle to another, is a progressive wave or travelling wave.
• Example: a sound wave, a pressure wave of compressions and rarefactions travelling along the direction of propagation.

Characteristics of Progressive Waves

• Each particle of the medium executes the same type of vibration — simple harmonic motion about its mean position.
• All vibrating particles have the same amplitude, period and frequency.
• The phase, that is the state of vibration, changes from one particle to the next.
• No particle remains permanently at rest; each comes to rest only momentarily, at the extreme positions of its vibration.
• Particles attain their maximum velocity while passing through their mean positions.
• Energy is transferred along the wave during propagation; there is no transfer of matter.
• The wave travels through the medium with a certain velocity, decided by the properties of the medium.
• Progressive waves are of two types: transverse and longitudinal.
• In a transverse wave the particles vibrate perpendicular to the direction of propagation, producing crests and troughs. In a longitudinal wave they vibrate along the direction of propagation, producing compressions and rarefactions.
• Both transverse and longitudinal mechanical waves can travel through solids, but only longitudinal waves can travel through fluids.

A question asking to "state the characteristics of progressive waves" for two marks wants any two of these, stated in full sentences.

Equation of a Progressive Wave

When a mechanical wave passes through an elastic medium, the displacement of the particle at position $x$ at time $t$ is

$$y(x,t) = f(x - vt)$$

The combination $(x - vt)$ appears because the disturbance produced at $x = 0$ at time $t$ reaches a point $x = x_c$ on the right at time $\left(t + \dfrac{x_c}{v}\right)$; equivalently, the disturbance of the particle at $x_c$ at time $t$ originated on the left at time $\left(t - \dfrac{x_c}{v}\right)$. So this represents a wave travelling in the positive $x$ direction with constant speed $v$.

The function $f$ depends on the motion of the source. If the source performs simple harmonic motion, the wave is a sine or cosine of $(x - vt)$, multiplied by a factor that makes the argument dimensionless:

$$y(x,t) = A\sin(kx - \omega t)$$

$$y(x,t) = A\sin(kx + \omega t) \qquad \text{(wave travelling along } -x)$$

where $A$ is the amplitude, $k = \dfrac{2\pi}{\lambda}$ is the wave number, $\lambda$ is the wavelength and $\omega$ is the angular frequency. The speed is $v = \dfrac{\omega}{k}$. SI units: $k$ in rad m$^{-1}$, $\lambda$ in m, $\omega$ in rad s$^{-1}$. If $T$ is the period, $n = \dfrac{1}{T} = \dfrac{\omega}{2\pi}$ is the frequency in Hz.

Two different forms — the board's own answer

Starting from $y(x,t) = A\sin(kx - \omega t)$ for a wave along the positive $X$ direction:

• Using $k = \dfrac{2\pi}{\lambda}$: $\quad y(x,t) = A\sin\left(\dfrac{2\pi x}{\lambda} - \omega t\right)$
• Using $\omega = 2\pi n$: $\quad y(x,t) = A\sin 2\pi\left(\dfrac{x}{\lambda} - nt\right)$
• Using $n = \dfrac{1}{T}$: $\quad y(x,t) = A\sin 2\pi\left(\dfrac{x}{\lambda} - \dfrac{t}{T}\right)$
• Using $v = n\lambda = \dfrac{\lambda}{T}$: $\quad y(x,t) = A\sin\dfrac{2\pi}{\lambda}(x - vt)$

Any two of these earn the marks. The same wave is equally often written as $y = A\sin 2\pi\left(nt - \dfrac{x}{\lambda}\right)$, which differs only by an overall sign, that is by a phase of $\pi$; this is the form used when comparing with a given numerical equation.

Phase Relations

$$\Delta\phi = \dfrac{2\pi}{\lambda}\,\Delta x \qquad \text{(two particles, same instant)}$$

$$\Delta\phi = 2\pi n\,\Delta t = \dfrac{2\pi}{T}\,\Delta t \qquad \text{(same particle, two instants)}$$

Standard Results

• Progressive wave: $y = A\sin(kx - \omega t)$ along $+x$; $y = A\sin(kx + \omega t)$ along $-x$.
• $k = \dfrac{2\pi}{\lambda}$, $\omega = 2\pi n$, $n = \dfrac{1}{T}$, $v = \dfrac{\omega}{k} = n\lambda$.
• To read constants off a given equation, first force it into the form $A\sin 2\pi\left(nt - \dfrac{x}{\lambda}\right)$: the coefficient of $t$ inside $2\pi(\ )$ is $n$, and the coefficient of $x$ is $\dfrac{1}{\lambda}$.