The diagram below shows three different modes of vibration P, Q and R of the same string of a given length.

(a) Which vibration will produce a louder sound and why?

(b) Which vibration will produce sound of maximum shrillness (or pitch) and why?

(c) What is the ratio of the wavelength of vibrations P and R?

(a) The vibration which produces loudest sound is R, because it's amplitude is maximum.

(b) The vibration which produces maximum shrillness (or pitch) is P, because it's frequency is maximum.

(c) Let l be the length of the string.

Wavelength of P (λ_p) = $\dfrac{2l}{3}$

Wavelength of R (λ_R) = 2l

Therefore, ratio of λ_p : λ_R =

$= \dfrac{2l}{3 \times 2l} \\[0.5em] = \dfrac{1}{3} \\[0.5em]$

Hence,

λ_P : λ_R = 1 : 3