Traveling Waves and Sound: An Introduction

Longitudinal Waves

In a longitudinal wave, the particles of the medium move parallel to the direction of the wave's travel. This motion creates regions of compression (where particles are crowded together) and expansion (where they are spread apart).

Sound waves are the primary example of longitudinal waves. When a loudspeaker vibrates, it creates compressions in the air that travel outward. In seismology, P waves (primary waves) are longitudinal waves that expand and compress the ground.

Mathematical Descriptions of Sinusoidal Waves

Many traveling waves follow a sinusoidal pattern, which is the same shape seen in simple harmonic motion (SHM). Because the motion of each individual particle in the medium is periodic, we use the same terminology to describe waves that we use for oscillations.

Key Wave Parameters

• Amplitude ($A$): The maximum value of displacement from the equilibrium position.
• Period ($T$): The time interval required to complete one full cycle of motion.
• Frequency ($f$): The number of cycles per unit time, defined by the relationship $f = 1/T$.
• Wavelength ($\lambda$): The distance spanned by one complete cycle of the wave.

Exam Tip: To distinguish between a "snapshot graph" and a "history graph," look at the axes. A snapshot graph shows displacement vs. position ($x$) at a fixed time, where you can measure wavelength. A history graph shows displacement vs. time ($t$) for a single point, where you can measure the period.

The Wave Function

To describe a traveling wave mathematically, we combine the displacement as a function of position and time into a single wave function $y(x, t)$:

$$y(x, t) = A \cos\left(2\pi\left(\frac{x}{\lambda} \mp \frac{t}{T}\right)\right)$$

Or, using angular wave number ($k = 2\pi/\lambda$) and angular frequency ($\omega = 2\pi f$):

$$y(x, t) = A \cos(kx \mp \omega t)$$

Note the sign convention: a wave traveling to the right (positive $x$ direction) uses a minus sign in the argument, while a wave traveling to the left uses a plus sign.

Wave Speed and the Medium

A fundamental principle of wave mechanics is that wave speed is a property of the medium, not the wave itself. The speed does not depend on the amplitude or the shape of the pulse.

Speed on a String

For a wave on a stretched string, the speed $v$ depends on the tension ($T_s$) and the linear mass density ($\mu$, which is mass per unit length):

$$v_{\text{string}} = \sqrt{\frac{T_s}{\mu}}$$

Increasing the tension makes the string respond more rapidly, increasing the speed. Conversely, increasing the linear density (making the string heavier) increases inertia, which decreases the speed.

Speed of Sound

In a gas, the speed of sound depends on the temperature ($T$) and the molar mass ($M$) of the gas:

$$v_{\text{sound}} = \sqrt{\frac{\gamma RT}{M}}$$

In air, the speed of sound increases as the temperature rises. For example, at $0^{\circ}\text{C}$, the speed is $331\text{ m/s}$, but at $20^{\circ}\text{C}$, it increases to $343\text{ m/s}$.

Comparison of Speeds

While mechanical waves are limited by their medium, electromagnetic waves (like light) do not require a medium and can travel through a vacuum at the constant speed $c = 3.00 \times 10^8\text{ m/s}$.

Table 15.1: Comparison of wave speeds.

Worked Example: Lightning and Thunder

Problem: During a thunderstorm, you see a flash of lightning. 8.0 seconds later, you hear the thunder. How far away was the lightning strike? (Assume the speed of sound in air is $343\text{ m/s}$ at $20^{\circ}\text{C}$).

Strategy: This involves two waves: light and sound. Because the speed of light is so high ($3.00 \times 10^8\text{ m/s}$), the time it takes for the light to reach you is negligible. Therefore, the 8.0-second delay is entirely due to the time it takes the sound wave to travel the distance.

Solution: Using the relationship between distance, speed, and time:

$$d = v \cdot \Delta t$$ $$d = (343\text{ m/s}) \cdot (8.0\text{ s})$$ $$d = 2744\text{ m} \approx 2.7\text{ km}$$

The lightning strike was approximately 2.7 km away.