(i) A wire of length 80 cm has a frequency of 256 Hz. Calculate the length of a similar wire under similar tension, which will have frequency 1024 Hz.

(ii) A certain sound has a frequency of 256 hertz and a wavelength of 1.3 m.

1. Calculate the speed with which this sound travels.

2. What difference would be felt by a listener between the above sound and another sound travelling at the same speed, but of wavelength 2.6 m?

(i) Given,

l₁ = 80 cm, f₁ = 256 Hz, f₂ = 1024 Hz, l₂ = ?

Since, f ∝ $\dfrac{1}{l}$

Therefore, f₁l₁ = f₂l₂

or

$f$2 = \dfrac{f1l1}{f2} \\[0.5em] = \dfrac{256 \times 80}{1024} \\[0.5em] = 20 \text{ cm}

(ii) Given, f = 256 Hz, λ = 1.3 m

(a) V = fλ = 256 x 1.3 = 332.8 m s^-1

(b) Speed of second sound (V') = 332.8 m s^-1

Wavelength of second sound (λ') = 2.6 m

$\text{f} = \dfrac{V'}{λ'} \\[0.5em] = \dfrac{332.8}{2.6} \\[0.5em] = 128 \text{ Hz}$

As frequency of second sound is less as compared to the first sound, hence, to the listener, the first sound of wavelength 1.3 m will appear to be shriller than the second sound of wavelength 2.6 m.