Question

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.

Short answer

a) The longest wavelength for waves travelling on the string if standing waves are to be set up is 240 cm

b) The second longest wavelength for waves travelling on the string if standing waves are to be set up is 120 cm

c) The third longest wavelength for waves travelling on the string if standing waves are to be set up is 80.0 cm

d) Sketches of the waves are drawn.

Step-by-step solution

Given data

Length of string, L = 120 cm or 1.20 cm

Understanding the concept of standing waves

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, $\lambda =2L/n.......\left(1\right)$

Step 3(a): Calculation of the longest wavelength for waves

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$240cm$

Step 4(b): Calculation of the second longest wavelength for waves

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

Step 5(c): Calculation of the third longest wavelength for waves

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$80.0cm$

Step 6(d): Sketching the graph of the waves

The above sketches represent wave nature at different integral numbers.