During propagation of a plane progressive mechanical wave,
The transverse displacement of a string (clamped at its both ends) is given by $y(x, t)=0.06 \sin \left(\frac{2 \pi x}{3}\right) \cos (120 \pi t)$. All the points on the string between two consecutive nodes vibrate with
A train, standing in a station yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with a speed of $10 \mathrm{~m} / \mathrm{s}$. Given that the speed of sound in still air is $340 \mathrm{~m} / \mathrm{s}$. Then
Which of the following statement are true for a stationary waves?
A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire?
Wire of twice the length vibrates in its second harmonic. Thus, if the tuning fork resonates at $L$, it will resonate at $2 L$. This can be explained as below The sonometer frequency is given by
$$\mathrm{v}=\frac{n}{2 L} \sqrt{\frac{T}{m}} \quad(n=\text { number of loops })$$
$$\begin{aligned} &\text { Now, as it vibrates with length } L \text {, we assume } v=v_1\\ &\begin{aligned} n & =n_1 \\ \therefore \quad v_1 & =\frac{n_1}{2 L} \sqrt{\frac{T}{m}} \quad \text{... (i)} \end{aligned}\end{aligned}$$
$$\begin{aligned} &\text { When length is doubled, then }\\ &v_2=\frac{n_2}{2 \times 2 L} \sqrt{\frac{T}{m}}\quad \text{.... (ii)} \end{aligned}$$
Dividing Eq. (i) by Eq. (ii), we get
$$\frac{v_1}{v_2}=\frac{n_1}{n_2} \times 2$$
To keep the resonance
$$\begin{aligned} & \frac{v_1}{v_2}=1=\frac{n_1}{n_2} \times 2 \\ & n_2=2 n_1 \end{aligned}$$
Hence, when the wire is doubled the number of loops also get doubled to produce the resonance. That is it resonates in second harmonic.