Sound and Waves Physics Lesson 17 by Owen Borville 12.11.2025
Sound is a disturbance of matter (pressure wave) that is transmitted from its source outward. Sound is one type of wave. Hearing is the perception of sound.
Sound can be modeled in terms of pressure or in terms of displacement of molecules. The human ear is sensitive to frequencies between 20 Hz and 20 kHz.
The relationship of the speed of sound, vw, its frequency, ƒ, and its wavelength λ is vw = ƒλ which is the same relationship for all waves.
The speed of sound depends on the medium and the state of the medium. In a fluid, because the absence of shear forces, sound waves are longitudinal. A solid can support both longitudinal and transverse sound waves.
Speed of sound in fluid is v = √B/ρ, where B is the bulk modulus or resistance to compression, and ρ is the density of the fluid.
Speed of sound in a solid is v = √Y/ρ, where Y is Young's modulus and ρ is the density of the fluid. Other elastic modulus values can be used for some solids.
In an ideal gas, the speed of sound is v = √γRT/M, where γ is the adiabatic index, R is the ideal gas constant, T is the absolute temperature, and M is the molar mass.
In air, the speed of sound is related to air temperature T by v = (331 m/s) √Tk/273K = 331 m/s √1+ Tc/273 degrees Celsius (vw is the same for all frequencies and wavelengths of sound in air).
Intensity is the same for a sound wave as for all waves: I = P/A, where P is the power crossing area A. SI units for I is watts per meter squared. Intensity of a sound wave is also related to the pressure amplitude Δp:
I = (Δp)^2/2ρv, where ρ is the density of the medium in which the sound wave travels and v is the speed of sound in the medium.
Sound intensity level in units of decibels (dB) is beta β (dB) = 10 log10(I/Io) where Io = 10^-12 W/m^2 is the threshold intensity of hearing.
The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer. The actual change in the frequency is called the Doppler shift.
A sonic boom is constructive interference of sound created by an object moving faster than sound. A sonic boom is a type of bow wake created when any wave source moves faster than the wave propagation speed.
For a stationary observer and a moving source, the observed frequency ƒobs = ƒs[vw/(vw +/- vs)] where ƒs is the frequency of the source, vs is the speed of the source, and vw is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away.
For a stationary source and moving observer, the observed frequency is ƒobs = ƒs[(vw +/-vobs)/vw], where vobs is the speed of the observer.
The doppler shift for the observed frequency is fobs = fs (v+/-vo)/(v+/-vs)
Sound interference and resonance have the same properties defined for all waves. In air columns, the lowest frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively they are called harmonics.
The resonant frequencies of a tube closed at one end are: ƒn = nvw/4L, n = 1, 3, 5, ..., ƒ1 is the fundamental and L is the length of the tube.
The resonant frequencies of a tube open at both ends are ƒn = nvw/2L, n = 1, 2, 3, ...
Resonant wavelengths of a tube closed at one end = λn = 4L/n, n = 1, 3, 5, ...
Resonant wavelengths of a tube open at both ends = λn = 2L/n, n = 1, 3, 5, ...
The range of audible frequencies is 20 to 20,000 Hz. Those sounds above 20,000 Hz are ultrasound, whereas those below 20 Hz are infrasound.
The perception of frequency is pitch. The perception of intensity is loudness. Loudness has units of phons.
Acoustic impedance is = Z = ρv, where ρ is the density of a medium through which the sound travels and v is the speed of sound through that medium.
The intensity reflection coefficient, a, a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the incident wave, is a = [(Z2 - Z1)^2]/[(Z1 + Z2)^2]. The intensity reflection coefficient is a unitless quantity.
Destructive interference can be used to reduce unwanted sound. Sound has the same properties of interference and resonance as defined for all waves.
Some musical instruments can be modeled as pipes that have symmetrical boundary conditions: open at both ends or closed at both ends. Other musical instruments can be modeled as pipes that have anti-symmetrical boundary conditions: closed at one end and open at the other.
Some instruments, such as the pipe organ, have several tubes with different lengths. Instruments such as the flute vary the length of the tube by closing the holes along the tube. The trombone varies the length of the tube using a sliding bar.
String instruments produce sound using a vibrating string with nodes at each end. The air around the string oscillates at the frequency of the string. The relationship for the frequencies for the string is the same as for the symmetrical boundary conditions of the pipe, with the length of the pipe replaced by the length of the string and the velocity replaced by v = √Ft/μ
Beats are created with a beat frequency that is equal to the absolute value of the difference in the frequencies when two sound waves that differ in frequency interfere.
Mach number is the velocity of a source divided by the speed of sound, M = vs/v. When a sound source moves faster than the speed of sound, a shock wave is produced as the sound waves interfere. A sonic boom is the intense sound that occurs as the shock wave moves along the ground. The angle the shock wave produces can be found as sin θ = v/vs = 1/M.
A bow wake is produced when an object moves faster than the speed of a mechanical wave in the medium, such as a boat moving through the water.
Sound is a disturbance of matter (pressure wave) that is transmitted from its source outward. Sound is one type of wave. Hearing is the perception of sound.
Sound can be modeled in terms of pressure or in terms of displacement of molecules. The human ear is sensitive to frequencies between 20 Hz and 20 kHz.
The relationship of the speed of sound, vw, its frequency, ƒ, and its wavelength λ is vw = ƒλ which is the same relationship for all waves.
The speed of sound depends on the medium and the state of the medium. In a fluid, because the absence of shear forces, sound waves are longitudinal. A solid can support both longitudinal and transverse sound waves.
Speed of sound in fluid is v = √B/ρ, where B is the bulk modulus or resistance to compression, and ρ is the density of the fluid.
Speed of sound in a solid is v = √Y/ρ, where Y is Young's modulus and ρ is the density of the fluid. Other elastic modulus values can be used for some solids.
In an ideal gas, the speed of sound is v = √γRT/M, where γ is the adiabatic index, R is the ideal gas constant, T is the absolute temperature, and M is the molar mass.
In air, the speed of sound is related to air temperature T by v = (331 m/s) √Tk/273K = 331 m/s √1+ Tc/273 degrees Celsius (vw is the same for all frequencies and wavelengths of sound in air).
Intensity is the same for a sound wave as for all waves: I = P/A, where P is the power crossing area A. SI units for I is watts per meter squared. Intensity of a sound wave is also related to the pressure amplitude Δp:
I = (Δp)^2/2ρv, where ρ is the density of the medium in which the sound wave travels and v is the speed of sound in the medium.
Sound intensity level in units of decibels (dB) is beta β (dB) = 10 log10(I/Io) where Io = 10^-12 W/m^2 is the threshold intensity of hearing.
The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer. The actual change in the frequency is called the Doppler shift.
A sonic boom is constructive interference of sound created by an object moving faster than sound. A sonic boom is a type of bow wake created when any wave source moves faster than the wave propagation speed.
For a stationary observer and a moving source, the observed frequency ƒobs = ƒs[vw/(vw +/- vs)] where ƒs is the frequency of the source, vs is the speed of the source, and vw is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away.
For a stationary source and moving observer, the observed frequency is ƒobs = ƒs[(vw +/-vobs)/vw], where vobs is the speed of the observer.
The doppler shift for the observed frequency is fobs = fs (v+/-vo)/(v+/-vs)
Sound interference and resonance have the same properties defined for all waves. In air columns, the lowest frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively they are called harmonics.
The resonant frequencies of a tube closed at one end are: ƒn = nvw/4L, n = 1, 3, 5, ..., ƒ1 is the fundamental and L is the length of the tube.
The resonant frequencies of a tube open at both ends are ƒn = nvw/2L, n = 1, 2, 3, ...
Resonant wavelengths of a tube closed at one end = λn = 4L/n, n = 1, 3, 5, ...
Resonant wavelengths of a tube open at both ends = λn = 2L/n, n = 1, 3, 5, ...
The range of audible frequencies is 20 to 20,000 Hz. Those sounds above 20,000 Hz are ultrasound, whereas those below 20 Hz are infrasound.
The perception of frequency is pitch. The perception of intensity is loudness. Loudness has units of phons.
Acoustic impedance is = Z = ρv, where ρ is the density of a medium through which the sound travels and v is the speed of sound through that medium.
The intensity reflection coefficient, a, a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the incident wave, is a = [(Z2 - Z1)^2]/[(Z1 + Z2)^2]. The intensity reflection coefficient is a unitless quantity.
Destructive interference can be used to reduce unwanted sound. Sound has the same properties of interference and resonance as defined for all waves.
Some musical instruments can be modeled as pipes that have symmetrical boundary conditions: open at both ends or closed at both ends. Other musical instruments can be modeled as pipes that have anti-symmetrical boundary conditions: closed at one end and open at the other.
Some instruments, such as the pipe organ, have several tubes with different lengths. Instruments such as the flute vary the length of the tube by closing the holes along the tube. The trombone varies the length of the tube using a sliding bar.
String instruments produce sound using a vibrating string with nodes at each end. The air around the string oscillates at the frequency of the string. The relationship for the frequencies for the string is the same as for the symmetrical boundary conditions of the pipe, with the length of the pipe replaced by the length of the string and the velocity replaced by v = √Ft/μ
Beats are created with a beat frequency that is equal to the absolute value of the difference in the frequencies when two sound waves that differ in frequency interfere.
Mach number is the velocity of a source divided by the speed of sound, M = vs/v. When a sound source moves faster than the speed of sound, a shock wave is produced as the sound waves interfere. A sonic boom is the intense sound that occurs as the shock wave moves along the ground. The angle the shock wave produces can be found as sin θ = v/vs = 1/M.
A bow wake is produced when an object moves faster than the speed of a mechanical wave in the medium, such as a boat moving through the water.