This question aims to find the effect on wave formulae when the frequency and tension in the string increase.
There are two formulae to calculate the waves on the string and these are:
\[ v = \lambda f \]
\[ v = \sqrt { \frac { T } { \mu }} \]
Here, v is the speed of the wave in the string, f represents the frequency of that wave, T is the tension produced in the string, and $ \mu $ represents the mass per unit length of the string. Considering a standard straight string having the mass and length both constant, we have to find the tension and frequency of that string.
Expert Answer
We can increase the tension in the string if we put the frequency constant in case 1 and we can calculate the effect of this increase in tension on the other variables used in the formulae like $ \lambda $, $ v $, $ f $, $ T $ and $ \mu $
Two weights are used to calculate the increase in tension of the spring. Two weights are suspended to the hook attached to the spring. The following effect on the variables occurred:
\[ v \propto T \]
According to the given expression of velocity and tension, the velocity is directly proportional to the tension in the string. If the velocity increases, the tension in the spring also increases.
$ \lambda $ represents the wavelength which is directly proportional to the tension in the string. The increase in one quantity causes an increase in another quantity.
\[ \mu = constant \]
Mass per unit length of the string will be constant as given in the question.
\[ f = constant \]
The frequency of the waves in the string will be constant as given.
The frequency of the waves in the string can be increased by changing the input frequency on the frequency generator and studying the effect of this frequency on the other variables used in the formulae like $ \lambda $, $ v $, $ f $, $ T $ and $ \mu $.
By changing the frequency:
\[ v \propto f \]
Velocity increases as the frequency increases because velocity is directly proportional to the frequency of the waves.
\[ f \propto \frac { 1 } { \lambda } \]
$ \lambda $ decreases with the increase in the frequency of the wave as it is inversely proportional to the frequency.
\[ \mu = constant \]
Mass per unit length of the string will be constant with the increase of the frequency as given in the question.
\[ T = constant \]
The tension in the string will be constant as given in the question.
Numerical Results
The increase in the tension causes an increase in wavelength and velocity while the increase in frequency causes a decrease in wavelength and an increase in velocity.
Example
Study the effect on the string if $ \lambda $ increases by keeping the frequency constant.
By changing the frequency:
\[ v \propto \lambda \]
Velocity increases as the wavelength increases because velocity is directly proportional to the wavelength of the waves.
\[ \lambda \propto \frac { 1 } { f } \]
$ \lambda $ increases with the decrease in the frequency of the wave as it is inversely proportional to the frequency.
\[ \mu = constant \]
Mass per unit length of the string will be constant with the increase of the frequency as given in the question.
\[ T = constant \]
The tension in the string will be constant as given in the question.