All Solutions
Page 504: Practice Problems
$$
f_3=3f_1=3(525.0Hz)=1575;Hz
$$
1575;Hz
$$
We know that the wavelength and frequency determine the speed of a wave. They are related by the following formula:
$$
begin{equation}
v = lambda cdot f
end{equation}
$$
We know that when a standing wave is formed in a bottle or pipe open at one end, the length of the pipe is equal to an odd number of wavelength fourths, that is:
$$
begin{align*}
L = frac{n cdot lambda}{4} , text{ for } , n = 1,3,5 tag{2}
end{align*}
$$
Since we know the fundamental frequency to be equal $f = 330text{ Hz}$, and the speed of sound in air to equal $v = 343 ; frac{text{m}}{text{s}}$ the wavelength of the wave can easily be found using formula (1) as follows:
$$
begin{align*}
v = f cdot lambda
end{align*}
$$
Rearranging
$$
begin{align*}
& lambda = frac{v}{f} \
& lambda = frac{343 ; frac{text{m}}{text{s}}}{330text{ Hz}} approx 1.0text{ m}
end{align*}
$$
Now for the length, we know the first harmonic is the one with maximal wavelength, for given pipe distance, knowing this we set the integer $n = 1$, and from formula (2) get:
$$
begin{align*}
L = frac{n cdot lambda}{4}
end{align*}
$$
Using the value for the wavelength we found we get
$$
begin{align*}
L = frac{1 cdot 1.0text{ m}}{4} = 0.25text{ m}
end{align*}
$$
$lambda_3=dfrac{4L}{3}$
Solve for L
$$
L=dfrac{3lambda_3}{4}=dfrac{3}{4}(0.22m)=0.17;m
$$
0.17;m
$$