| 521 |
speed of sound in a medium.
$B=$ bulk
modulus of medium,
$\rho=$ density
of medium.
Note that for a solid bar,
Young's modulus would be
used instead
|
\[
v=\sqrt{\frac{B}{\rho}}
\]
|
| 521 |
speed of sound in a solid bar.
$Y=$ Young's
modulus of the bar
|
\[
v=\sqrt{\frac{Y}{\rho}}
\]
|
| 521 |
general form of speed of mechanical waves
|
\[
v=\sqrt{\frac{\mathrm{elastic\ property}}{\mathrm{inertial\ property}}}
\]
|
| 521 |
speed of sound in air.
$T_\mathrm{C}=$ temperature
of air in degrees Celsius.
|
\[
v=\left(331\ \mathrm{m}/\mathrm{s}\right)\sqrt{1+\frac{T_C}{273^\circ C}}
\]
|
| 521 |
speed of sound in air
at $0^\circ C, 32^\circ F.$
|
\[
v=331\ \mathrm{m}/\mathrm{s}
\]
|
| 521 |
speed of sound in air
at $20^\circ C,68^\circ F.$
(Use this value when working problems involving
$v=$ speed
of sound.)
|
\[
v=343\ \mathrm{m}/\mathrm{s}
\]
|
| 523 |
displacement wave function of a sinusoidal sound wave.
$s_{\mathrm{max}}=$ maximum
displacement amplitude of
a volume element from equilibrium
|
\[
s\left(x,t\right)=s_{\mathrm{max}}\cos{\left(kx-\omega t\right)}
\]
|
| 523 |
pressure variation wave function of a sinusoidal sound wave.
Note that the pressure wave is $90^\circ=\frac{\pi}{2}$
out of phase with the displacement wave, i.e.
$k$ and $\omega$ are the same in both equations.
|
\[
\Delta P=\Delta P_{\mathrm{max}}\sin{\left(kx-\omega t\right)}
\]
|
| 524 |
pressure amplitude.
maximum change in pressure from
equilibrium of a sinusoidal sound wave. $v=$ speed of
sound in air, $\rho=$ density of air.
|
\[
\Delta P_{\mathrm{max}}=\rho v\omega s_{\mathrm{max}}
\]
|
| 524 |
displacement amplitude,
maximum displacement of a sinusoidal sound wave.
|
\[
s_{\mathrm{max}}=\frac{\Delta P_{\mathrm{max}}}{\rho v\omega}
\]
|
| 506 |
displacement speed of transverse wave (simple harmonic motion)
|
\[
v_s=\frac{\partial y}{\partial t}=-\omega A\sin{\left(kx-\omega t\right)}
\]
|
| 506 |
displacement acceleration of transverse wave (simple harmonic motion)
|
\[
v_s=\frac{\partial v_y}{\partial t}=-\omega^2A\cos{\left(kx-\omega t\right)}
\]
|
| 506 |
maximum displacement speed (simple harmonic motion).
Occurs when $y=0.$
|
\[
v_{s,\mathrm{max}}=-\omega A
\]
|
| 506 |
maximum displacement acceleration (simple harmonic motion).
Occurs when $y=\pm A.$
|
\[
a_{s,\mathrm{max}}=-\omega^2A
\]
|
| 525 |
total kinetic energy in one wavelength of a sound wave
i.e. of a displacement wave $s\left(x,t\right)$ of air.
|
\[
K_\lambda=\frac{1}{4}\rho A\left(\omega s_{\mathrm{max}}\right)^2\lambda
\]
|
| 525 |
total potential energy in one wavelength of a sound wave
i.e. of a displacement wave $s\left(x,t\right)$ of air.
|
\[
U_\lambda=K_\lambda
\]
|
| 526 |
total mechanical energy in one wavelength of a sound wave
i.e. of a displacement wave $s\left(x,t\right)$ of air.
|
\[
E_\lambda=K_\lambda+U_\lambda=\frac{1}{2}\rho A\left(\omega s_{\mathrm{max}}\right)^2\lambda
\]
|
| 526 |
power delivered by a sound wave.
$v=$ speed
of sound in air,
$\rho=$ density
of air,
$A=$ cross-sectional
area of moving volume of air,
$\omega=$ angular
frequency of $s\left(x,t\right),$
wave function for gas displacement
|
\[
\mathscr{P}=\frac{1}{2}\rho Av\left(\omega s_{\mathrm{max}}\right)^2
\]
|
| 526 |
intensity.
We define the
intensity $I$ of a wave,
or the
power per unit area,
to be the rate at which
the energy being transported by the wave flows through
a unit area $A$ perpendicular to the direction of travel
of the wave. From the equation we see that a periodic
sound wave is proportional to the square of the
displacement amplitude and to the square of the
angular frequency.
|
\[
I=\frac{\mathscr{P}}{A}=\frac{1}{2}\rho v\left(\omega s_{\mathrm{max}}\right)^2=\frac{\Delta P_{\mathrm{max}}^2}{2\rho v}
\]
|
| 526 |
power delivered by a sound wave in terms of intensity and area
|
\[
\mathscr{P}=IA
\]
|
| 527 |
sound level.
$I=$ intensity,
$\beta=$ sound
level (decibels).
|
\[
\beta=10\log{\left(\frac{I}{I_0}\right)},\ \ I_0=1.00\times{10}^{-12}\frac{\mathrm{W}}{\mathrm{m}^2}
\]
|
| 527 |
combined sound level.
Sound levels don't add, i.e. if two
sources of sound are experienced at a point, with $\beta_1$ and
$\beta_2$ the sound levels from each source separately, then the
combined sound level $\beta\neq\beta_1+\beta_2.$ Intensities do
add, however, giving a way to find the combined sound level.
|
\[
\beta=10\log{\left(\frac{\sum I}{I_0}\right)},\ \ I_0=1.00\times{10}^{-12}\frac{\mathrm{W} }{\mathrm{m}^2}
\]
|
| 527 |
threshold of hearing
|
\[
I_0=1.00\times{10}^{-12}\frac{\mathrm{W} }{\mathrm{m}^2}
\]
|
| 527 |
threshold of pain
|
\[
I=1.00\frac{\mathrm{W}}{\mathrm{m}^2}
\]
|
| 528 |
wave intensity at distance $r$ from the source of a spherical wave.
The intensity is the same at all points on a given wave front of a spherical wave.
|
\[
I=\frac{\mathscr{P}_{\mathrm{av}}}{A}=\frac{\mathscr{P}_{\mathrm{av}}}{4\pi r^2}
\]
|
| 528 |
relation between
intensity,
amplitude,
and
radius
for a
spherical wave
|
\[
\frac{I_1}{I_2}=\frac{s_1^2}{s_2^2}=\frac{r_2^2}{r_1^2}
\]
|
| 529 |
wave function for an outgoing spherical wave.
$\frac{s_0}{r}=s_{\mathrm{max}}$ is the maximum
displacement amplitude,
$s_0=$ displacement
amplitude at unit distance is constant and
characterizes the whole wave
|
\[
\psi\left(r,t\right)=\frac{s_0}{r}\sin{\left(kr-\omega t\right)}
\]
|
| 529 |
wave function for a plane wave
perpendicular to the
$x$-axis
and
traveling in the $x$ direction. The intensity is
the same at all points on a given wave front of a
plane wave.
|
\[
\psi\left(x,t\right)=A\sin{\left(kx-\omega t\right)}
\]
|
| 533 |
observed frequency due to the doppler effect.
$f=$ true
frequency,
$v=$ speed
of sound,
$v_O=$ speed
of observer,
$v_S=$ speed
of source of sound
waves. $+v_O$ is used when
the observer moves toward the source, $-v_O$
away from the source. $-v_S$ is used when the
source moves toward the observer, $+v_S$ away
from the observer. $f^\prime\gt f$ when the
net of their motion is such that the observer
and source are moving toward each other.
$f^\prime\lt f$ when they are moving away from
each other. That is, a net increase in relative
velocity corresponds to an increase in relative
frequency, a net decrease in relative velocity,
a decrease in relative frequency.
|
\[
f^\prime=\frac{v\pm v_O}{v\mp v_S}f
\]
|
|
observed wavelength for a source moving toward
a (moving or stationary) observer.
|
\[
\lambda^\prime=\lambda-\frac{v_S}{f}
\]
|
|
observed wavelength for a source moving
toward a (moving or stationary) observer.
|
\[
\lambda^\prime=\lambda+\frac{v_S}{f}
\]
|
| 533 |
observed wavelength for a source moving
toward or away from a stationary observer.
|
\[
\lambda^\prime=\frac{v}{f^\prime}
\]
|
| 533 |
observed wavelength for an observer
moving toward a stationary source.
The wavelength doesn't change in this case.
|
\[
\lambda^\prime=\lambda
\]
|
| 535 |
apex half-angle of the conical envelope (shock wave)
produced by a source traveling with speed greater
than the speed of sound $\left(v_S\gt v\right).$
|
\[
\sin{\theta=\frac{v}{v_S}}
\]
|
| 535 |
radius of the spherical wave at time $t$
produced at time $t_n$ by a source moving
with speed $v_S\gt v.$
The center
of the wave is the point where the source was located at
time $t_n.$
|
\[
r_n=v\left(t-t_n\right)
\]
|
| 534 |
distance traveled by a source at time $t$
moving with speed $v_S.$
|
\[
d=v_St
\]
|
| 535 |
mach number.
For example, mach 2, occurs
when the source moves at twice the
speed of sound, i.e. $v_S=2v.$
|
\[
\frac{v_S}{v}
\]
|
Alex Notes