The Nature of Sound and Waves

The mathematical relationship is expressed as:

$$c = \sqrt{\frac{K}{\rho}}$$

This formula reveals an interesting tension: increasing density ($\rho$) actually tends to slow sound down, while increasing stiffness ($K$) speeds it up. This explains why sound travels much faster in steel than in air; although steel is much denser than air, it is significantly stiffer, and the increase in stiffness outweighs the increase in density.

Speed of Sound in Air

In the atmosphere, the number of molecules available to carry the wave depends on the atmospheric pressure (the weight of the atmosphere acting on a surface). Temperature also plays a critical role. As temperature increases, the density of air decreases, which affects the speed. For air at standard pressure (1 atm), the speed can be calculated using the following formula:

$$c_{air} = 331.45 + (0.6 \cdot T)$$

where $T$ is the temperature in degrees Celsius.

Exam Tip: For most general calculations, if a specific temperature isn't provided, it is common to assume a speed of sound of approximately $340\text{ m/s}$, which corresponds to roughly $15^\circ\text{C}$ (room temperature).

Wave Properties: Wavelength and Frequency

Every sound wave can be described by its physical dimensions and its timing.

Wavelength ($\lambda$) is the physical distance in space occupied by one full cycle of the wave. In a sound wave, this is the distance between two consecutive compressions or two consecutive rarefactions.

Frequency ($f$) refers to how often these cycles occur per second, measured in Hertz (Hz). These two properties are linked to the speed of sound by the following equation:

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

One vital property to remember is that the speed of sound does not depend on the frequency. Whether you are listening to a low-pitched bass note or a high-pitched whistle, the waves will arrive at your ears at the same time.

Wavefronts and Sound Intensity

As a sound wave moves away from its source, it forms a wavefront—an imaginary line connecting all the points of the wave that are currently experiencing the same state of disturbance. The shape of this wavefront depends on the source:

• Plane waves have flat wavefronts.
• Spherical waves expand outward like an inflating bubble.

Sound Pressure vs. Sound Intensity

It is important to distinguish between how we measure the "strength" of a sound.

Sound Pressure ($p$) refers to the pressure changes caused by the vibration of the source. It is measured in Pascals (Pa) or $\text{N/m}^2$. It can be calculated by the force applied over a specific area:

$$p = \frac{F}{A}$$

Sound Intensity ($I$) describes the amount of sound power ($P$) traveling through a specific area of the wavefront. For a plane wave, intensity is:

$$I = \frac{P}{A}$$

For a spherical source (like a tuning fork), the power spreads out over the surface of an expanding sphere ($A = 4\pi r^2$). This leads to the Inverse Square Law:

$$I = \frac{P}{4\pi r^2}$$

The Inverse Square Law tells us that as you move away from a source, the intensity drops significantly. Specifically, for every doubling of the distance from the source, the intensity decreases by $6\text{ dB}$.

Summary Comparison

• Sound Pressure indicates how much the particles are being compressed or rarefied.
• Sound Intensity indicates how much energy/power is actually being transferred through the medium.