a. What are the three longest wavelengths for standing waves on a 60 cm long string that is fixed at both ends?

Key Concepts

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In a fixed medium, such as a string, this results in specific points called nodes (where there is no movement) and antinodes (where the movement is maximum). The pattern of these waves is determined by the length of the string and the wavelength of the waves.

Wavelength and Frequency Relationship

The wavelength of a wave is the distance between successive crests or troughs, and it is inversely related to frequency. For standing waves on a string, the wavelength can be calculated using the formula λ = 2L/n, where L is the length of the string and n is the mode number (an integer representing the harmonic). This relationship is crucial for determining the wavelengths of standing waves.

Harmonics

Harmonics refer to the different modes of vibration that can occur in a string fixed at both ends. The fundamental frequency (first harmonic) has the longest wavelength, while higher harmonics (second, third, etc.) have shorter wavelengths. The wavelengths for these harmonics can be calculated using the formula mentioned earlier, allowing us to find the three longest wavelengths for a given string length.