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Page 416: Practice Problems
$f = 440, mathrm{Hz}$
$T = 20, mathrm{^oC}$
$dfrac{lambda}{2} = ?$
Since we have to determine resonance spacing, following equation:
$$
lambda = dfrac{v}{f}
$$
becomes:
$$
dfrac{lambda}{2} = dfrac{v}{2f}
$$
When we put known values into the previous equation we get:
$$
dfrac{lambda}{2} = dfrac{343, mathrm{m/s}}{2 cdot 440, mathrm{Hz}}
$$
$$
boxed{dfrac{lambda}{2} = 0.39, mathrm{m}}
$$
dfrac{lambda}{2} = 0.39, mathrm{m}
$$
$f = 440, mathrm{Hz}$
$dfrac{lambda}{2} = 110, mathrm{cm} = 1.1, mathrm{m} rightarrow lambda = 2.2, mathrm{m}$
$v = ?$
Speed is given by:
$$
v = lambda f
$$
When we put known values into the previous equation we get:
$$
v = 2.2, mathrm{m} cdot 440, mathrm{Hz}
$$
$$
boxed{v = 970, mathrm{m/s}}
$$
v = 970, mathrm{m/s}
$$
$T = 27, mathrm{^oC}$
$dfrac{lambda}{2} = 20.2, mathrm{cm} = 0.202, mathrm{m} rightarrow lambda = 0.404, mathrm{m}$
$v = 347, mathrm{m/s}$
$f = ?$
Frequency is given by:
$$
f = dfrac{v}{lambda}
$$
When we out known values into the previous equation we get:
$$
f = dfrac{347, mathrm{m/s}}{0.404, mathrm{m}}
$$
$$
boxed{f = 859, mathrm{Hz}}
$$
f = 859, mathrm{Hz}
$$
The longest wavelength is:
$lambda_0 = (2) * (2.65) = 5.3 m$
The lowest frequency is:
$f_0 = dfrac{v}{lambda_0} = dfrac{343}{5.3} = 64.717 = 64.7 Hz$
b)
$f_1 = 2 f_0 = (2)*(64.717) = 129 Hz$
$$
f_2 = 3 f_0 = (3)*(64.717) = 194 Hz
$$
b) $129 Hz$ , $194 Hz$
$$lambda=2cdot Ltag1$$
From that we get the value for $lambda$. The value is $5.3~mathrm{m}$. Now we will insert that value in equation for frequency which is:
$$f=frac{v}{lambda}tag2$$
$$begin{align*}
f&=frac{v}{lambda}\
&=frac{343~mathrm{frac{cancel{m}}{s}}}{5.3~mathrm{cancel{m}}}\\
&=boxed{64.7~mathrm{frac{m}{s}}}
end{align*}$$
That is the lowest frequency.
$$begin{align*}
lambda_{1}&=L\
lambda_{2}&=frac{2}{3}L
end{align*}$$
Using the equation (2) we get:
$$begin{align*}
f_{1}&=frac{v}{lambda_{1}}\\
&=frac{343~mathrm{frac{cancel{m}}{s}}}{2.65~mathrm{cancel{m}}}\\
&=boxed{129.4~mathrm{frac{m}{s}}}end{align*}$$
Now the same for the second wavelength:
$$begin{align*}
f_{2}&=frac{v}{lambda_{2}}\
&frac{3cdot 343~mathrm{frac{cancel{m}}{s}}}{5.3~mathrm{cancel{m}}}\\
&=boxed{194.1~mathrm{frac{m}{s}}}
end{align*}$$
Those are the values for two resonant frequencies.
&text{a)}~ f= 64.7~mathrm{frac{m}{s}}\
&text{b)}~f_{1}=129.4~mathrm{frac{m}{s}} ~text{and}~f_{1}=194.1~mathrm{frac{m}{s}}
end{align*}
$$
