Plane Waves in Fluid Media

🎯 Learning Objectives:

The simplest form of sound propagation in a continuous medium is the plane wave. This is the direct solution of the one-dimensional sound propagation problem introduced earlier. Here, we derive important quantities such as speed of sound, sound intensity, energy density, and the sound pressure level.

The Plane Wave Solution

The one-dimensional acoustic wave equation derived previously is:

This partial differential equation admits a general solution known as the d’Alembert solution:

where: $f(x - ct)$ represents a wave traveling in the +x direction, $g(x + ct)$ represents a wave traveling in the –x direction, and $c$ is the speed of sound in the medium.
The form $(x - ct)$ indicates that the waveform shape remains constant but moves along the x-axis with speed $c$. For simplicity, consider a plane progressive wave moving only in the positive x-direction, i.e., $g = 0$:

This represents the propagation of constant-phase surfaces (planes) perpendicular to the x-axis. All points in planes parallel to the y–z plane experience identical pressure variations, defining a plane wave.

Harmonic Plane Waves

In many acoustic applications, sources are sinusoidal in time. Therefore, we assume a harmonic plane wave:

where:
$p_m$ is the pressure amplitude,
$\omega = 2\pi f$ is the angular frequency,
$k = \frac{\omega}{c}$ is the wave number,
and $c$ is the speed of sound.

The instantaneous particle velocity $v(x,t)$ is related to the pressure by the linearized Euler equation:

Substituting the pressure expression and integrating with respect to time gives:

This shows that pressure and particle velocity are in phase for a plane progressive wave in an inviscid fluid. The ratio between them defines the characteristic acoustic impedance:

Speed of Sound in Gases

For an ideal gas, the speed of sound is obtained from thermodynamic relations:

where:
$\kappa = \frac{C_p}{C_v}$ is the adiabatic index,
$p_0$ and $\rho_0$ are the equilibrium pressure and density.

At normal conditions (air, $\theta = 20^\circ \mathrm{C}$):

The temperature dependence can be estimated by:

where $\theta$ is the air temperature in °C.

Speed of Sound in Liquids

Although liquids are often approximated as incompressible, small pressure perturbations propagate as acoustic waves due to their finite adiabatic compressibility:

The speed of sound then becomes:

For example, in water at 25°C, $c \approx 1497\ \text{m/s}$, which is roughly 4.4 times that in air.

Characteristic Acoustic Impedance

The characteristic impedance $Z_0 = \rho_0 c$ represents the medium’s resistance to sound propagation. It defines the ratio of sound pressure to particle velocity in a plane progressive wave.
Typical values include:

Large impedance mismatches cause strong reflections at boundaries between materials, such as air–water or air–metal interfaces.

Energy and Intensity of Plane Waves

The instantaneous intensity (acoustic power per unit area) is given by:

For a harmonic plane wave:

The time-averaged intensity over one period $T$ is:

where;
$p_{\text{rms}} = \frac{p_m}{\sqrt{2}}$ is the root-mean-square sound pressure.

The energy density $E$ is composed of potential (pressure-related) and kinetic (motion-related) terms:

For a harmonic plane wave, these two components are equal on average, leading to:

Thus, the average intensity and average energy density are related by:

Complex Representation of Plane Waves

It is convenient to use complex notation (phasor representation) for harmonic waves:

with the complex amplitude:

The particle velocity becomes:

This formalism simplifies the analysis of reflection, transmission, and interference, especially when multiple waves overlap.

Physical Interpretation

• A plane wave is an idealized model valid when the wavefront curvature is negligible, such as in regions far from small sources or near the axis of a large aperture.
• The pressure and velocity are in phase, resulting in a net flow of acoustic energy.
• The wave impedance determines how efficiently energy transfers between media.
• Real fluids exhibit small losses (viscosity, heat conduction), leading to attenuation, which can be introduced later through complex wave numbers.

Wide-band Waves and Signals

Fourier Representation of Acoustic Signals

From the harmonic wave, we can directly construct any other wave function or, referring to one measurement point in space, a time function of sound pressure, called an acoustic signal.
The process of combining multiple harmonic signals to form a more complex waveform is based on the principle of superposition and is mathematically described using the Fourier transformation.

For a continuous periodic pressure–time function with period $T_0$, the transformation into the frequency domain yields a set of Fourier coefficients $\bar{S}_m$, which represent the complex amplitudes of the respective harmonic components.
The collection of these complex amplitudes constitutes the spectrum of the signal.

The inverse process, or Fourier synthesis, reconstructs the original time-domain pressure signal $p(t)$ by summing all its harmonic components:

Periodic and Aperiodic Signals

For periodic signals, the spectrum consists of discrete frequency components located at integer multiples of the fundamental frequency $f_0$.
Such a spectrum is called a line spectrum, and its frequency resolution is given by:

In contrast, when the signal becomes aperiodic (i.e., $T_0 \to \infty$), the spacing between frequency lines tends to zero, and the spectrum becomes continuous.
This leads to the representation of non-repetitive or transient signals in terms of a continuous distribution of frequencies.

Broadband and Narrowband Sounds

When a spectrum contains a wide range of frequency components, the sound is referred to as broadband or wideband sound.
Examples include white noise and impulsive sounds.

In contrast, if the spectrum contains only a few discrete frequency components or a narrow frequency band, the signal is classified as narrowband - as in the case of a pure tone or single-frequency harmonic wave.

Sound Level and Energy

The range of sound pressures in air must be understood in relation to our initial assumption - that acoustic waves are small perturbations (fluctuations in pressure, density, or particle velocity) superimposed on static equilibrium conditions.
At sea level, the static atmospheric pressure is approximately $p_0 = 100 \ \text{kPa}$.
However, the sound pressures associated with audible acoustic waves are several orders of magnitude smaller, typically ranging between:

• The lower limit ($10^{-5} \ \text{Pa}$) corresponds roughly to the threshold of hearing, the minimum pressure variation perceivable by the human ear.
• The upper limit (around $10^3 \ \text{Pa}$) corresponds to the threshold of pain, beyond which sound becomes physically intolerable.

Thus, acoustic pressure amplitudes are indeed many orders of magnitude lower than the static atmospheric pressure $p_0$, validating the linear acoustics assumption used in plane wave theory.

The Logarithmic (Decibel) Scale

In practical acoustics, the enormous dynamic range of sound pressures is mapped onto a logarithmic scale - the decibel scale - which compresses the range of values between approximately 0 dB (threshold of hearing) and 130 dB (threshold of pain).
This scaling reflects how the human auditory system perceives loudness differences — roughly proportional to ratios, not absolute differences, in sound pressure.

The sound pressure level (SPL) is defined as:

where,

• $\tilde{p}$ = root mean square (RMS) sound pressure
• $p_{\text{ref}} = 20 \ \mu \text{Pa}$ = reference sound pressure, approximately the threshold of human hearing at mid frequencies
• $L_p$ = sound pressure level, measured in decibels (dB)

The RMS pressure is defined by:

Energy Density and Wave Relationships

In a plane progressive sound wave, the total acoustic energy density $w$ (energy per unit volume) can be expressed equivalently in terms of either particle velocity or sound pressure:

where,

• $\rho_0$ = mean density of the medium
• $v$ = instantaneous particle velocity
• $p$ = instantaneous sound pressure
• $c$ = speed of sound in the medium

At instants of maximum particle velocity, the energy is entirely kinetic, while at zero velocity, it is entirely potential. Over one complete cycle, the average total energy is the sum of both contributions.

Example Reference Levels

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