Sound

Sound is generated by vibrations that create oscillating pressure waves within a medium. These pressure waves travel parallel to the direction of propagation, making sound a longitudinal wave. Sound cannot be produced or transmitted in a vacuum since it requires a material medium.

• When the frequency of vibrations is below human hearing range, it is known as infrasound
• when the frequency is above human hearing range, it is called ultrasound.

Ultrasound utilizes the reflection property of sound waves to create images in diagnostic applications. In an ultrasound imaging system, a source emits these high-frequency waves, which travel through a medium and reflect off internal structures. The reflected waves are then captured by a detector, allowing for the construction of an image. Although sound also exhibits refraction and diffraction, ultrasound imaging primarily relies on reflection to provide detailed visualizations of internal tissues and organs.

Relative speed of sound in solids, liquids, and gases

The speed of sound is highest in solids because their high stiffness allows pressure waves to propagate quickly, even though they are also typically denser.

In liquids, the speed is lower, as they are less stiff but still less compressible than gases.

In gases, despite their low density, the high compressibility significantly slows the transmission of sound.

Additionally, under similar conditions, sound travels faster in hotter media than in colder media.

Intensity of sound, decibel units, log scale

The intensity of sound is defined as the power per unit area and is measured in $W/m^2$.

This intensity is often expressed on a logarithmic scale in decibels using the formula

$$\beta =10\log (I/I_0)$$

where $I$ is the measured intensity and $I_0$ is the reference intensity, typically $10^{-12}W/m^2$.

For example, a sound with intensity equal to $I_0$ registers 0 dB; 10 times $I_0$ gives 10 dB; 100 times $I_0$ corresponds to 20 dB; and 1000 times $I_0$ results in 30 dB. This logarithmic scale reflects the fact that human perception of loudness increases much more gradually than the linear increase in intensity.

As sound travels through a medium, it undergoes attenuation, meaning its intensity gradually decreases, especially in materials that are soft, elastic, viscous, or less dense.

Doppler Effect: moving sound source or observer, reflection of sound from a moving object

The Doppler effect explains how the observed frequency of a wave changes due to the relative motion between the source and the observer:

• When the source moves toward a stationary observer, the observed frequency increases, as described by the equation $fo=fs(v/(v-vs))$, where $fs$ is the actual frequency, $v$ is the speed of sound, and $vs$ is the velocity of the source.
• Conversely, if the observer moves toward a stationary source, the observed frequency becomes $fo=fs((v+vo)/v)$, with $vo$ representing the observer’s speed.
• When both the source and observer approach each other, the observed frequency is given by $fo=fs((v+vo)/(v-vs))$.
• If the source moves away from a stationary observer, the **observed frequency **decreases to $fo=fs(v/(v+vs))$. Similarly, if the observer recedes from a stationary source, $fo=fs((v-vo)/v)$.
• In situations where one moves toward while the other moves away, the observed frequency may be either higher or lower than the actual frequency—for example, if the source is approaching but the observer is receding, $fo=fs((v-vo)/(v-vs))$, and if the source recedes while the observer approaches, $fo=fs((v+vo)/(v+vs))$.

Pitch, harmonics and resonance in pipes and strings

Pitch is the human perception of the frequency of sound; higher frequencies produce higher pitches. This relationship is central to the resonance observed in musical instruments like pipes and strings.

The frequency of a sound is determined by the equation

$$f=v/\lambda$$

where $f$ represents frequency, $v$ is the speed of sound, and $\lambda$ denotes the wavelength.

• In instruments with two open ends, such as many string instruments and open pipes, the length (L) is related to the wavelength by $L=n/2\lambda$, where n is an integer representing the harmonic mode.
• In pipes with one closed end, the resonant condition is given by $L=(2n–1)/4\lambda$.

Harmonics

Harmonics are the distinct frequencies produced by a vibrating system that are integer multiples of its fundamental frequency. The lowest of these frequencies, known as the first harmonic ($n=1$), establishes the basic pitch. The next frequency, called the second harmonic ($n=2$), is exactly twice the fundamental frequency, and further harmonics follow as higher integer multiples. These additional frequencies contribute to the overall sound quality or timbre by enriching the original tone.

Shock waves

Shock waves are abrupt disturbances in pressure and density that occur when an object travels faster than the speed of sound. In this situation, the sound waves produced by the object merge into a single, intense front because they cannot propagate ahead of it. This concentrated wave, which is perceived as a sonic boom, represents a sharp discontinuity in the medium and marks the boundary between high-pressure and low-pressure regions.

This results in a loud sonic boom due to constructive interference of waves along the cone’s surface, while destructive interference inside the cone leads to reduced sound intensity.