Waves and Sound II: Energy, Intensity, Decibels, and Standing Waves

Energy and Intensity in Mechanical Waves

Introduction to Mechanical Waves

Mechanical waves are disturbances that propagate through a medium, transporting energy but not matter. As these waves move, they carry the energy provided by their source through the medium. Understanding how energy is distributed and measured in waves is essential for analyzing sound and other wave phenomena.

Circular, Spherical, and Plane Waves

• Circular Waves: These waves move outward in two dimensions, such as ripples on a pond's surface. The wave crests form concentric circles, each spaced one wavelength apart.

• Spherical Waves: These waves propagate in three dimensions, like sound radiating from a point source. The energy spreads over the surface of expanding spheres.

• Plane Waves: At large distances from the source, both circular and spherical waves can be approximated as plane waves, where the wave fronts appear flat and parallel.

Power, Energy, and Intensity

The power (P) of a wave is the rate at which energy is transported by the wave. Intensity (I) is defined as the power transmitted per unit area perpendicular to the direction of energy flow. For a spherical wave, the intensity decreases with the square of the distance from the source because the energy spreads over a larger area.

• General Intensity Formula: $I = \frac{P}{a}$

• Intensity for Spherical Waves: $I = \frac{P}{4\pi r^2}$

Example: If a plane, circular, and spherical wave all start with the same intensity and travel the same distance, the plane wave maintains its intensity, while the circular and spherical waves' intensities decrease due to spreading.

Loudness of Sound and the Decibel Scale

Loudness and Human Perception

Loudness is the subjective perception of sound intensity. The human ear can detect sounds with intensities from $10^{-12}$ W/m2 (threshold of hearing, $I_0$) up to about 10 W/m2 (threshold of pain). The perceived loudness is not directly proportional to intensity; a tenfold increase in intensity is typically perceived as only about twice as loud.

• Threshold of Hearing: $I_0 = 1.0 \times 10^{-12}$ W/m2

The Decibel Scale

Because the range of audible intensities is so large, sound levels are measured on a logarithmic scale called the decibel (dB) scale. The sound level β in decibels is given by:

• $\beta = (10\ \mathrm{dB}) \log_{10}\left(\frac{I}{I_0}\right)$

• To find intensity from sound level: $I = I_0 10^{\beta/(10\ \mathrm{dB})}$

Table: Typical Sound Levels and Intensities

Sound	β (dB)	I (W/m2)
Threshold of pain	130	10
Rock concert	120	1.0
Home stereo at max	110	0.10
Pneumatic hammer (2 m)	100	0.010
Niagara Falls (viewpoint)	90	1.0 × 10−3
Vacuum cleaner	80	1.0 × 10−4
Busy traffic	70	1.0 × 10−5
Normal conversation (1 m)	60	1.0 × 10−6
Quiet restaurant	50	1.0 × 10−7
Residential street	40	1.0 × 10−8
Classroom during test	30	1.0 × 10−9
Whisper (1 m)	20	1.0 × 10−10
Person breathing (3 m)	10	1.0 × 10−11
Threshold of hearing	0	1.0 × 10−12

The Principle of Superposition and Interference

Superposition Principle

When two or more waves pass through the same region of space, the resulting displacement is the algebraic sum of the individual displacements. This is known as the principle of superposition.

Constructive and Destructive Interference

• Constructive Interference: Occurs when waves combine to produce a displacement greater than either wave alone.

• Destructive Interference: Occurs when waves combine to produce a smaller displacement, possibly canceling each other out.

Standing Waves

Formation and Properties of Standing Waves

A standing wave forms when two sinusoidal waves of equal amplitude and wavelength travel in opposite directions and interfere. Standing waves do not appear to travel; instead, certain points (nodes) remain stationary, while others (antinodes) oscillate with maximum amplitude.

• Nodes: Points of destructive interference where the medium does not move.

• Antinodes: Points of constructive interference with maximum oscillation.

Mathematical Description of Standing Waves

• The displacement of a sinusoidal traveling wave is given by: $y(x, t) = A \cos \left(2\pi \left(\frac{x}{\lambda} \pm \frac{t}{T}\right)\right)$ where $A$ is amplitude, $\lambda$ is wavelength, $T$ is period, and the sign indicates direction.

Example: Determining Wavelength from Standing Wave Patterns

Given a standing wave with a total length of 3.0 m and three loops (antinodes), the wavelength can be determined by noting that each loop corresponds to half a wavelength. Thus, $3\lambda/2 = 3.0$ m, so $\lambda = 2.0$ m.

Equation Summary

Concept	Equation or Description
Intensity (general)	$I = \frac{P}{a}$
Intensity (spherical wave)	$I = \frac{P}{4\pi r^2}$
Threshold of hearing	$I_0 = 1.0 \times 10^{-12}$ W/m2
Sound level (dB)	$\beta = (10\ \mathrm{dB}) \log_{10}\left(\frac{I}{I_0}\right)$
Intensity from sound level	$I = I_0 10^{\beta/(10\ \mathrm{dB})}$
Displacement of sinusoidal wave	$y(x, t) = A \cos \left(2\pi \left(\frac{x}{\lambda} \pm \frac{t}{T}\right)\right)$