Sound is produced by vibrations that create oscillating pressure waves in a material medium (such as air, water, or a solid). These pressure changes move parallel to the direction the wave travels, so sound is a longitudinal wave. Sound can’t be produced or transmitted in a vacuum because there’s no medium to carry the pressure variations.

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

Ultrasound imaging mainly uses the reflection of sound waves to form pictures of internal structures. A source emits high-frequency waves that travel through the body and reflect off boundaries between tissues. A detector (often the same probe) receives the reflected waves, and the system uses their timing and strength to build an image. Although sound can also refract and diffract, diagnostic ultrasound relies primarily on reflection to produce detailed views of tissues and organs.

Relative speed of sound in solids, liquids, and gases

The speed of sound depends largely on how stiff (hard to compress) a material is.

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 the power per unit area carried by the wave, measured in $W/m^2$.

Because intensities can span a huge range, we often use a logarithmic scale in decibels (dB):

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

where:

• $I$ is the measured intensity
• $I_0$ 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 matches the idea 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. Attenuation is especially strong 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 describes how the observed frequency of a wave changes when there is 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 how you perceive the frequency of a sound: higher frequency means higher pitch. Musical instruments use resonance to reinforce certain frequencies, which is why pipes and strings produce clear notes.

Frequency, wave speed, and wavelength are related by:

$$f=v/\lambda$$

where:

• $f$ is the frequency
• $v$ is the speed of sound in the medium
• $\lambda$ is the wavelength

Resonance happens when the instrument’s length matches a standing-wave pattern.

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

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 ($n=1$), sets the basic pitch. The second harmonic ($n=2$) is twice the fundamental frequency, and higher harmonics continue as larger integer multiples. These additional frequencies shape the sound’s quality (timbre) by adding overtones to the fundamental.

Shock waves

Shock waves are abrupt changes in pressure and density that occur when an object moves faster than the speed of sound. Because the object is outrunning its own sound waves, the waves can’t spread out in front of it. Instead, they pile up into a single, intense wavefront. This concentrated front is heard as a sonic boom and marks a sharp boundary in the medium between higher-pressure and lower-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.