Question

Using special techniques called string harmonics (or "flageolet tones"), stringed instruments can produce the first few overtones of the harmonic series. While a violinist is playing some of these harmonics for us, we take a picture of the vibrating string (see figures). Using an oscilloscope, we find the violinist plays a note with frequency f = 805 Hz in figure (a).

Part (a) How many nodes does the standing wave in figure (a) have?

Part (b) How many antinodes does the standing wave in figure (a) have?

Part (c) The string length of a violin is about L = 33 cm. What is the wavelength of the standing wave in figure (a) in meters?

Part (d) The fundamental frequency is the lowest frequency that a string can vibrate at (see figure (b)). What is the fundamental frequency for our violin in Hz?

Part (e) In terms of the fundamental frequency f1, what is the frequency of the note the violinist is playing in figure (c)?

Part (f) Write a general expression for the frequency of any note the violinist can play in this manner, in terms of the fundamental frequency f1 and the n, the number of antinodes on a standing wave.

Part (g) What is the frequency, in hertz, of the note the violinist is playing in figure (d)?

Answer 1

(a) The standing wave in figure (a) has 2 nodes.

(b) The standing wave in figure (a) has 1 antinode.

(c) The wavelength of the standing wave in figure (a) is approximately λ = 0.825 m.

(d) The fundamental frequency for the violin is f₁ = 242.4 Hz.

(e) The frequency of the note the violinist is playing in figure (c) is 3 times the fundamental frequency, so f = 3f₁ = 727.2 Hz.

(f) The general expression for the frequency of any note the violinist can play is f = nf₁, where n is the number of antinodes on a standing wave.

(g) The frequency of the note the violinist is playing in figure (d) is 4 times the fundamental frequency, so f = 4f₁ = 969.6 Hz.

Step-by-step explanation:

In figure (a), the standing wave has two nodes (points of zero displacement) and one antinode (point of maximum displacement). The nodes and antinode are characteristic features of standing waves. In figure (b), where the fundamental frequency is depicted, the standing wave has one node and two antinodes.

To calculate the wavelength (λ) in figure (a), we use the formula λ = 2L/n, where L is the string length and n is the number of nodes. Substituting the values (L = 0.33 m, n = 2), we find λ = 0.825 m.

The fundamental frequency (f₁) of the violin is calculated using the formula f₁ = v/2L, where v is the wave velocity (speed of sound). Given the string length L = 0.33 m and the velocity of sound in air v ≈ 343 m/s, we get f₁ = 242.4 Hz.

For the note in figure (c), it has three antinodes, indicating a frequency of 3f₁ = 727.2 Hz. The general expression for any note played by the violinist is f = nf₁, where n is the number of antinodes. Finally, in figure (d), with four antinodes, the frequency is 4f₁ = 969.6 Hz.