14.1 Speed of Sound, Frequency, and Wavelength

Sound waves are vibrations that travel through air, water, and solids. They're characterized by amplitude, frequency, wavelength, and speed. These properties determine how loud or high-pitched a sound is and how fast it moves through a given medium.

The speed of sound depends on what it's traveling through. It moves faster in stiffer, more elastic materials. Temperature also affects sound speed in gases. Understanding these factors is essential for predicting how sound behaves in different environments and for solving wave equation problems.

Sound Wave Characteristics and Behavior

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Characteristics of sound waves

Sound waves propagate as longitudinal waves. That means the particles in the medium vibrate parallel to the direction the wave travels, creating alternating compressions (high-pressure regions) and rarefactions (low-pressure regions).

A few key points about sound waves:

• They require a medium to propagate. Sound cannot travel through a vacuum because there are no particles to compress and expand.
• They transfer energy through the medium without transferring matter. The particles themselves just oscillate back and forth around their equilibrium positions.

Four properties define any sound wave:

• Amplitude represents the maximum displacement of particles from equilibrium. Greater amplitude means a louder sound.
• Frequency ($f$), measured in Hertz (Hz), is the number of complete wave cycles passing a fixed point per second. Higher frequency means higher pitch.
• Wavelength ($\lambda$), measured in meters (m), is the distance between two consecutive compressions or two consecutive rarefactions.
• Speed ($v$), measured in meters per second (m/s), is the rate at which the wave propagates through the medium.

Factors affecting sound speed

The speed of sound depends on two properties of the medium:

• Elasticity (bulk modulus, $B$) measures how strongly a material resists compression. A stiffer material snaps back faster when compressed, so sound travels faster through it.
• Density ($\rho$) is the mass per unit volume. For a given elasticity, a denser material has more inertia per unit volume, which slows the wave down.

These two factors combine in the equation:

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

A common misconception: students see that solids are denser than gases and assume sound should be slower in solids. But solids are also far more elastic (much higher $B$), and that effect wins out. The elasticity increase more than compensates for the density increase.

Speed of sound in gases:

• Depends on temperature. Higher temperatures mean faster-moving molecules, which transmit compressions more quickly.
• At 20°C, the speed of sound in air is approximately 343 m/s (about 1,125 ft/s).

Speed of sound in liquids and solids:

• Generally much faster than in gases because their bulk modulus is so much larger.
• In water at 20°C, sound travels at about 1,482 m/s.
• In steel, sound travels at roughly 5,960 m/s.

Speed, frequency, and wavelength relationships

The three wave properties are connected by the wave equation:

$v = f\lambda$

where $v$ is in m/s, $f$ is in Hz, and $\lambda$ is in meters. This equation works for all waves, not just sound.

You can rearrange it depending on what you need to solve for:

Finding wavelength:

$\lambda = \frac{v}{f}$

Example: A tuning fork vibrates at 440 Hz (the note A4) in air at 20°C. What's the wavelength?

$\lambda = \frac{343 \text{ m/s}}{440 \text{ Hz}} \approx 0.78 \text{ m}$

Finding frequency:

$f = \frac{v}{\lambda}$

Example: A sound wave in water has a wavelength of 1.5 m. What's its frequency?

$f = \frac{1482 \text{ m/s}}{1.5 \text{ m}} \approx 988 \text{ Hz}$

Notice that the same frequency sound has a much longer wavelength in water than in air, because the wave speed is so much higher. When a sound wave crosses from one medium to another, its frequency stays the same (set by the source), but its wavelength changes to satisfy $v = f\lambda$ in the new medium.

Wave propagation and interaction

Sound waves propagate by creating alternating regions of high and low pressure. Each particle pushes on its neighbor, transferring energy forward through the medium without any individual particle traveling far from its resting position.

The physical properties of the medium (density, elasticity, temperature) determine how efficiently this energy transfer happens. That's why the same sound source produces waves that travel at very different speeds in air, water, and steel.

Resonance occurs when an object is driven to vibrate at its natural frequency. At resonance, each new push from the driving force arrives in sync with the object's existing vibration, causing the amplitude to build up significantly. You'll see this concept become important when studying standing waves and musical instruments later in this unit.