Question

What are the three longest wavelengths for standing waves on a 240-cm-long string that is fixed at both ends?

Answer 1

For a standing wave if both ends are fixed, the wavelength must be such that the length of the string be an exact multiple of a half wavelength.
The longest wavelength must be such that the length of the string must be equal to half a wavelength, and therefore the wavelength must be double the length of the string; That is 240× 2 = 480 cm
The second longest wavelength must be such that the length of the string must be equal to a whole wavelength, so the second longest wavelength must be 240 cm.
The third longest wavelength must be such that the length of the string must be equal to 1.5 times the wavelength, so the wavelength must be 240/1.5 = 160 cm.