In a standing wave on a string with fixed ends, n represents the number of half-wavelengths that fit between the ends. Which option correctly states this?

In a standing wave on a string with fixed ends, n represents the number of half-wavelengths that fit between the ends. Which option correctly states this?

• A. The number of full wavelengths between ends.
• B. The number of nodes in the string.
• C. The number of half-wavelengths that fit between the ends.
• D. The speed of the wave.

Answer: (C)

In a string fixed at both ends, standing waves must fit an integer number of half-wavelengths between the ends. Each segment between adjacent nodes spans half a wavelength, so the length of the string equals n times λ/2, or L = n(λ/2). This makes λ = 2L/n and shows that n counts how many half-wavelength sections fit along the length. That’s why n is described as the number of half-wavelengths that fit between the ends. This differs from the number of full wavelengths between ends (that would be L/λ, not the mode number) and from the total number of nodes (which is n+1, counting both ends). The wave speed, determined by tension and linear density, doesn't depend on n. So the statement that n represents the number of half-wavelengths that fit between the ends is the best description.

In a string fixed at both ends, standing waves must fit an integer number of half-wavelengths between the ends. Each segment between adjacent nodes spans half a wavelength, so the length of the string equals n times λ/2, or L = n(λ/2). This makes λ = 2L/n and shows that n counts how many half-wavelength sections fit along the length. That’s why n is described as the number of half-wavelengths that fit between the ends.

This differs from the number of full wavelengths between ends (that would be L/λ, not the mode number) and from the total number of nodes (which is n+1, counting both ends). The wave speed, determined by tension and linear density, doesn't depend on n. So the statement that n represents the number of half-wavelengths that fit between the ends is the best description.