Question 1

What is a standing wave on a string?

Accepted Answer

A standing wave forms when equal-frequency, equal-amplitude waves travel in opposite directions and superpose. The nodes remain at zero displacement, the antinodes oscillate with maximum amplitude, and energy is exchanged locally between adjacent loops rather than transported along the string.

Question 2

What are the allowed frequencies of a string fixed at both ends?

Accepted Answer

The allowed normal-mode frequencies are f_n = nv/(2L), where n = 1, 2, 3 and so on. The corresponding wavelength is lambda_n = 2L/n. Thus the fundamental is f_1 = v/(2L).

Question 3

Why is the midpoint an antinode in the fundamental mode?

Accepted Answer

In the fundamental mode n = 1, the string forms one half-wavelength between the clamps. Therefore x = 0 and x = L are nodes, while x = L/2 is the only antinode. There is no node at the midpoint in this mode.

Question 4

How is the beat frequency produced?

Accepted Answer

Nearly equal but different frequencies make their relative phase change continuously. Their superposition then produces repeating maxima and minima in loudness or amplitude. The beat rate is the absolute frequency difference, |f1 - f2|, not their average.

Question 5

What does a beat rate of 4 beats per second mean?

Accepted Answer

It means four loudness maxima occur in one second, so successive maxima are approximately 0.25 seconds apart. For 220 Hz and 224 Hz sources, the exact beat rate is 4 beats s^-1.

Standing Waves & Beats — Two Waves, One String

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