Wave Characteristics and Speed

Displacement Relation In A Progressive Wave ($ y(x,t) = A\sin(kx - \omega t + \phi) $)

A progressive wave (or travelling wave) is a wave that transfers energy from one point to another in a medium or space. We can describe the displacement of a particle in the medium (for mechanical waves) or the value of the wave field (for electromagnetic waves) at a specific position and time using a mathematical function.

For a one-dimensional progressive harmonic wave travelling in the positive x-direction, the displacement $y$ of a particle at position $x$ and time $t$ can be described by a sinusoidal function:

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

where:

• $y$ is the displacement from the equilibrium position (for mechanical waves), or the value of the wave field.
• $A$ is the amplitude.
• $k$ is the angular wave number (or propagation constant).
• $\omega$ is the angular frequency.
• $t$ is time.
• $\phi$ is the initial phase constant.

The term $(kx - \omega t + \phi)$ is called the phase of the wave at position $x$ and time $t$.

For a wave travelling in the negative x-direction, the displacement relation is typically written as $ y(x,t) = A \sin(kx + \omega t + \phi) $. The sign between the $kx$ and $\omega t$ terms determines the direction of propagation.

The functional form $y(x,t) = f(x \pm vt)$ represents any progressive wave travelling with speed $v$, where $f$ is any function. For a harmonic wave, $f$ is a sine or cosine function.

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Amplitude And Phase

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Amplitude (A)

The amplitude ($A$) of a wave is the maximum displacement or magnitude of the wave variable from its equilibrium value. In the equation $y(x,t) = A \sin(kx - \omega t + \phi)$, the amplitude is the coefficient $A$.

• For a transverse wave on a string, $A$ is the maximum vertical displacement of a point on the string from its horizontal equilibrium position.
• For a longitudinal sound wave, $A$ can represent the maximum displacement of a particle from its equilibrium position, or the maximum variation in pressure or density from the average value (pressure amplitude or density amplitude).
• For an electromagnetic wave, $A$ represents the maximum strength of the electric or magnetic field.

The amplitude is related to the energy carried by the wave; typically, the energy is proportional to the square of the amplitude ($E \propto A^2$).

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Phase ($kx - \omega t + \phi$)

The phase of a wave at a particular position and time, denoted by $\Psi(x,t) = kx - \omega t + \phi$, determines the instantaneous state of oscillation of the particle or field at that point. It is an angle, measured in radians.

• All points on the wave with the same phase value are in the same state of oscillation (same displacement and velocity/field value and direction).
• A phase difference of $2\pi$ corresponds to one full wavelength or one full period.
• The term $\phi$ is the initial phase constant (or initial phase angle). It determines the state of oscillation at $x=0$ and $t=0$.
• The phase of a wave changes with both position ($x$) and time ($t$). Moving to a different position or waiting for some time will change the phase, indicating a different point in the oscillation cycle.

The concept of phase is crucial when dealing with phenomena like interference, where the relative phase of different waves determines whether they constructively or destructively interfere.

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Wavelength And Angular Wave Number ($ k = 2\pi/\lambda $)

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Wavelength ($\lambda$)

The wavelength ($\lambda$) is the spatial period of the wave. It is the shortest distance between two points on a wave that are in phase (i.e., points that are at the same point in their oscillation cycle, such as two successive crests, troughs, compressions, or rarefactions).

In the equation $y(x,t) = A \sin(kx - \omega t + \phi)$, if we fix the time $t$ and look at the wave's shape in space, the pattern repeats over a distance $\lambda$. The phase $(kx - \omega t + \phi)$ must change by $2\pi$ for the displacement to repeat. Let $x$ and $x+\lambda$ be two points at the same time $t$ with the same phase (differing by $2\pi$):

$ (k(x+\lambda) - \omega t + \phi) - (kx - \omega t + \phi) = 2\pi $

$ kx + k\lambda - \omega t + \phi - kx + \omega t - \phi = 2\pi $

$ k\lambda = 2\pi $

$ \lambda = \frac{2\pi}{k} $

Units of wavelength are metres (m).

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Angular Wave Number ($k$)

The angular wave number ($k$), also known as the propagation constant, is related to the wavelength. It is defined as $k = 2\pi/\lambda$. It represents the spatial frequency of the wave in radians per unit length.

Units of angular wave number are radians per metre (rad/m) or simply per metre (m$^{-1}$) since radians are dimensionless.

$k$ represents how the phase changes with distance at a fixed time.

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Period, Angular Frequency And Frequency ($ \omega = 2\pi\nu = 2\pi/T $)

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Period (T)

The period ($T$) is the temporal period of the wave. It is the minimum time interval after which the oscillation of a particle at a fixed position repeats itself.

In the equation $y(x,t) = A \sin(kx - \omega t + \phi)$, if we fix the position $x$ and look at how the displacement changes with time, the pattern repeats over a time $T$. The phase $(kx - \omega t + \phi)$ must change by $2\pi$ for the displacement to repeat. Let $t$ and $t+T$ be two instants at the same position $x$ with the same phase (differing by $2\pi$):

$ (kx - \omega (t+T) + \phi) - (kx - \omega t + \phi) = -2\pi $ (assuming $\omega$ is positive for a wave travelling in +x, so phase decreases with time)

$ kx - \omega t - \omega T + \phi - kx + \omega t - \phi = -2\pi $

$ -\omega T = -2\pi $

$ \omega T = 2\pi $

$ T = \frac{2\pi}{\omega} $

Units of period are seconds (s).

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Angular Frequency ($\omega$)

The angular frequency ($\omega$) is related to the period. It is defined as $\omega = 2\pi/T$. It represents the temporal frequency of the oscillation in radians per unit time. Units are radians per second (rad/s) or simply per second (s$^{-1}$). $\omega$ represents how the phase changes with time at a fixed position.

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Frequency ($\nu$ or f)

The frequency ($\nu$ or $f$) is the number of complete oscillations per unit time. It is the reciprocal of the period, $\nu = 1/T$. Units are Hertz (Hz) or s$^{-1}$.

The relationship between angular frequency and frequency is:

$ \omega = 2\pi \nu $

Combining the relationships: $ \omega = \frac{2\pi}{T} = 2\pi\nu $. This confirms the subheading formula.

The Speed Of A Travelling Wave ($ v = \omega/k = \nu\lambda $)

The speed of a travelling wave ($v$) is the rate at which a point of constant phase (like a crest or a compression) propagates through the medium or space. It is also called the phase velocity.

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Derivation from Wave Parameters

Consider the phase of a travelling wave: $\Psi(x,t) = kx - \omega t + \phi$. For a point of constant phase, $\Psi(x,t) = \text{constant}$.

As time $t$ increases, the position $x$ must change in a specific way for the phase to remain constant. Differentiating the phase with respect to time, holding the phase constant ($d\Psi = 0$):

$ d(kx - \omega t + \phi) = 0 $

$ k \, dx - \omega \, dt + 0 = 0 $

$ k \, dx = \omega \, dt $

The speed of the wave is the rate at which this point of constant phase moves, $v = dx/dt$.

$ v = \frac{dx}{dt} = \frac{\omega}{k} $

This is a fundamental relationship between the wave speed, angular frequency, and angular wave number.

We can also express the speed in terms of frequency ($\nu = \omega/(2\pi)$) and wavelength ($\lambda = 2\pi/k$ or $k = 2\pi/\lambda$):

$ v = \frac{\omega}{k} = \frac{2\pi\nu}{2\pi/\lambda} = \frac{2\pi\nu}{1} \times \frac{\lambda}{2\pi} = \nu\lambda $

The wave speed $v$ is equal to the product of its frequency $\nu$ and wavelength $\lambda$. This is a very commonly used formula for wave speed.

$ \mathbf{v = \nu\lambda} $

The speed of a wave is a property of the medium through which it travels (or space, for EM waves) and the type of wave, but it is generally independent of the wave's source, amplitude, frequency, or wavelength (as long as the amplitude is not too large or the frequency is within the medium's linear response range).

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Speed Of A Transverse Wave On Stretched String ($ v = \sqrt{T/\mu} $)

For a mechanical wave, the speed is determined by the medium's elastic properties and its inertia (density).

For a transverse wave travelling on a stretched string, the restoring force comes from the tension in the string, and the inertia is determined by the mass per unit length of the string. The speed of the transverse wave is given by:

$ v = \sqrt{\frac{T}{\mu}} $

where:

• $T$ is the tension in the string (in Newtons, N).
• $\mu$ is the linear mass density of the string (mass per unit length, in kg/m).

This formula shows that the wave speed increases with increasing tension (stronger restoring force) and decreases with increasing mass per unit length (greater inertia). This is consistent with intuition.

This speed is the speed at which transverse disturbances propagate along the string. It is independent of the frequency or amplitude of the wave (for ideal strings and small amplitudes).

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Speed Of A Longitudinal Wave (Speed Of Sound) ($ v = \sqrt{B/\rho} $ or $ \sqrt{Y/\rho} $)

For a longitudinal wave travelling through a medium, the restoring force is related to the medium's resistance to compression or expansion (elastic modulus), and the inertia is related to its density.

• For a longitudinal wave in a fluid (liquid or gas), the relevant elastic property is the Bulk modulus ($B$), which measures the resistance to volume change. The speed of the longitudinal wave (speed of sound) is:

  $ v = \sqrt{\frac{B}{\rho}} $

  where $\rho$ is the density of the fluid.
• For a longitudinal wave in a solid rod, the relevant elastic property is Young's modulus ($Y$), which measures the resistance to stretching or compression. The speed of the longitudinal wave is:

  $ v = \sqrt{\frac{Y}{\rho}} $

  where $\rho$ is the density of the solid. For longitudinal waves in bulk solids, other elastic constants are involved.

As discussed in the previous section, for gases, the Bulk modulus depends on the process (isothermal or adiabatic). For sound waves (rapid compressions/rarefactions), the process is approximately adiabatic, and $B_{adiabatic} = \gamma P$, so $v_{sound, gas} = \sqrt{\gamma P/\rho}$.

These formulas relate the speed of longitudinal waves (sound waves) to the material properties of the medium. They confirm that sound travels faster in stiffer media (higher $B$ or $Y$) and slower in denser media (higher $\rho$), and generally faster in solids than in liquids, and faster in liquids than in gases.

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