547 
resultant wave function
for two waves traveling to the right with the
same frequency, wavelength and amplitude, but
differing in phase.

\[
\begin{align*}
y &=y_1+y_2
=2A\cos{\left(\frac{\phi}{2}\right)}
\sin{\left(kx\omega t+\frac{\phi}{2}\right)}\\
&\text{where}\\
y_1 &=A\sin{\left(kx\omega t\right)}\\
y_2 &=A\sin{\left(kx\omega t+\phi\right)}
\end{align*}
\]

549 
path difference $\Delta r$
and
phase difference $\phi$
for two sound waves with identical wavelengths, amplitudes
and frequencies, following different paths $r_1$ and $r_2$
from a common origin to a common receiver, where
$\Delta r=\leftr_2r_1\right.$ The value of $\phi$ is
the phase difference at the point of the receiver where
the waves come back together.

\[
\Delta r=\frac{\phi}{2\pi}\lambda
\]

549 
constructive interference,
path difference
and
phase difference
when waves are in phase.
Either condition must hold for constructive
interference. The path difference implies the
phase difference, but not vice versa. The
intensity of the resultant wave is at a maximum.

\[
\begin{align*}
&\phi=2n\pi=0,2\pi,4\pi,\ldots\\
&\Delta r=\left(2n\right)\frac{\lambda}{2}=n\lambda=0,\lambda,2\lambda,3\lambda,4\lambda,\ldots
\end{align*}
\]

549 
destructive interference,
path difference and
phase difference when waves are "maximally"
out of phase.
Either condition must hold for
destructive interference. The path difference
implies the phase difference, but not vice versa.
The intensity of the resultant wave is at a minimum.

\[
\begin{align*}
&\phi=\left(2n+1\right)\pi=\pi,3\pi,5\pi,\ldots\\
&\Delta r=\left(2n+1\right)\frac{\lambda}{2}=\frac{\lambda}{2},\frac{3\lambda}{2},\frac{5\lambda}{2},\ldots
\end{align*}
\]

550 
wave function of a standing wave.
A standing wave is an oscillation
pattern with a stationary outline
that results from the superposition
of two identical waves traveling in
opposite directions. Because $y$ does
not contain a function of $kx\omega t,$
a standing wave is not a traveling wave.
$2A\sin{kx}=$ amplitude
of the particle at position $x$ in
the standing wave.
$2A=$ amplitude
of the standing wave.

\[
\begin{align*}
y &=y_1+y_2=\left(2A\sin{kx}\right)\cos{\omega t},\\
&\text{where}\\
y_1 &=A\sin{\left(kx\omega t\right)}\\
y_2 &=A\sin{\left(kx\omega t\right)}
\end{align*}
\]

551 
position of nodes in a standing wave.
Nodes occur at multiples of one half wavelength.
The distance between adjacent nodes
is $\frac{\lambda}{2}.$

\[
x=n\frac{\lambda}{2},n=0,1,2,3,\ldots
\]

551 
position of antinodes in a standing wave.
Antinodes occur at odd multiples of one quarter
wavelength. the distance between adjacent
antinodes is $\frac{\lambda}{2}.$ The distance
between a node and antinode is $\frac{\lambda}{4}.$

\[
x=\left(2n1\right)\frac{\lambda}{4},n=1,3,5,\ldots
\]

554 
natural frequencies of vibration of a taut string fixed
at both ends and a column of air inside a pipe open at
both ends.
$L=$ length
of string or pipe,
$T=$ tension
in string,
$\mu=$ linear
mass density of string. This equation implies that to
change the frequency of a string, one must change either
its tension or its length. natural frequencies form a
harmonic series

\[
\begin{align*}
&f_n=\frac{v}{\lambda_n}=\frac{n}{2L}v=\frac{n}{2L}\sqrt{\frac{T}{\mu}},n=1,2,3,\ldots\\
&\lambda_n=\frac{2L}{n},n=1,2,3,\ldots\\
&f_1,2f_1,3f_1,\ldots\\
&\lambda_1,\frac{1}{2}\lambda_1,\frac{1}{3}\lambda_1,\ldots
\end{align*}
\]

561 
natural frequencies of oscillation of a column of
air inside a pipe open at one end and closed at the
other.
(Only the odd harmonics are present.) natural
frequencies form a
harmonic series

\[
\begin{align*}
&f_{2n1}=\frac{v}{2\lambda_{2n1}}=\frac{2n1}{4L}v,n=1,2,3,\ldots\\
&\lambda_{2n1}=\frac{4L}{2n1},n=1,2,3,\ldots\\
&f_1,3f_1,5f_1,\ldots\\
&\lambda_1,\frac{1}{3}\lambda_1,\frac{1}{5}\lambda_1,\ldots
\end{align*}
\]

565 
resultant of two waves of equal
amplitude but different frequencies
at a point $x=0.$
The frequency is the average of the frequencies
(the argument of the second cosine without the $t$)
and the amplitude is the entire expression in square
brackets. Note that the amplitude and thus intensity
of the resultant wave varies with time.

\[
\begin{align*}
y &=\left[2A\cos{2\pi\left(\frac{f_1f_2}{2}\right)t}\right]\cos{2\pi\left(\frac{f_1+f_2}{2}\right)t}\\
&\text{where}\\
y_1 &=A\cos{\omega_1t}=A\cos{2\pi f_1t}\\
y_2 &=A\cos{\omega_2t}=A\cos{2\pi f_2t}
\end{align*}
\]

564566 
beat frequency.
Beating is the periodic variation in intensity
at a given point due to the superposition of two
waves having slightly different frequencies.

\[
f_b=\leftf_1f_2\right
\]
