Consider a wave on a string with constant tension. if the frequency of the wave is doubled, by what multiplicative factor does a) the speed and b) the wavelength change?

A) The multiplicative factor by which the speed of the wave changes when the frequency is doubled is; multiplicative factor of 2.

B) The multiplicative factor by which the wavelength of the wave changes when the frequency is doubled is; multiplicative factor of ½.

The formula for speed of a wave is given as;

v = fλ

Where;

v = speed

f = frequency

λ = wavelength

A) We are told that the frequency is doubled. This means f is now 2f. Thus;

v = (2f) × λ

v = 2(fλ)

Thus, it means the speed is doubled by a multiplicative factor of 2.

B) Similarly, let's make the wavelength the subject when the frequency is doubled.

v = (2f) × λ

λ = v/(2f)

λ = ½(v/f)

Thus,the wavelength changes by a multiplicative factor of ½.

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Explanation:

It is given that, a wave on a string with constant tension. The frequency of a wave and its speed is related by the following relation as :

............(1)

Where

v is the speed of the wave

l is the length of the string

(a) It is clear from equation (1) that the frequency of a wave is directly proportional to its speed. If the frequency of the wave is doubled, the speed of the wave also becomes double i.e. the speed increases by a factor of 2.

(b) The relationship between the frequency and the wavelength is given by :

It is clear from the above formula that the frequency is inversely proportional to the wavelength of the wave. If the frequency of a wave is doubled, the wavelength becomes half.

Hence, this is the required solution.

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