A 120 mlength of string is stretched between fixed supports. What are the (a) longest, (b) second longest, and (c) third longest wavelength for waves traveling on the string if standing waves are to be set up? (d)Sketch those standing waves.

Length of string, L = 120 cm or 1.20 cm

For standing waves, the distance between two adjacent nodes is half the wavelength. We know the relation between wavelength and the number of loops, using these relations we can find the longest, second longest, and third-longest wavelength for waves traveling on the string if standing waves are to be set up.

Formula:

The wavelength of oscillation for n-loops, $\lambda =2L/n.......\left(1\right)$

For longest wavelength n = 1 the wavelength for waves using equation (1) is given as:

$$\begin{gathered} \lambda =\frac{2\left(120cm\right)}{1} \\ =240cm \end{gathered}$$

Therefore, the longest wavelength for waves travelling on the string if standing waves are to be set up is$240cm$

For second longest wavelength n = 2 the wavelength for waves using equation (1) is given as:

$$\begin{gathered} \lambda =\frac{2\left(120cm\right)}{2} \\ =120cm \end{gathered}$$

Therefore, the second longest wavelength for waves travelling on the string if standing waves are to be set up is 120 cm

For third longest wavelength n = 3 the wavelength for waves using equation (1) is given as:

$$\begin{gathered} \lambda =\frac{2\left(120cm\right)}{3} \\ =80.0cm \end{gathered}$$

Therefore, the third longest wavelength for waves travelling on the string if standing waves are to be set up is$80.0cm$

The above sketches represent wave nature at different integral numbers.