FIGURE EX.21.8 shows a standing wave that is oscillating at...

FIGURE EX.21.8 shows a standing wave that is oscillating at frequency $f_{0}$ a. How many antinodes will there be if the frequency is doubled to $2 / 8 ?$ Explain. b. If the tension in the string is increased by a factor of four, for what frequency, in terms of $f_{0},$ will the string continue to oscillate as a standing wave with three antinodes? (FIGURE CAN'T COPY)

FIGURE EX.21.8 shows a standing wave that is oscillating at frequency $f_{0}$
a. How many antinodes will there be if the frequency is doubled to $2 / 8 ?$ Explain.
b. If the tension in the string is increased by a factor of four, for what frequency, in terms of $f_{0},$ will the string continue to oscillate as a standing wave with three antinodes?
(FIGURE CAN'T COPY)

Key Concepts

Standing Waves

Standing waves result from the interference of two waves traveling in opposite directions, leading to a fixed pattern of nodes, where the medium remains at rest, and antinodes, where the oscillation amplitude is maximized. This phenomenon is central to understanding vibrations in musical instruments, strings, and resonant cavities.

Normal Modes and Resonance

Normal modes refer to the specific patterns of standing waves that can be sustained by a system when it is driven at its resonant frequencies. Each mode corresponds to a unique distribution of nodes and antinodes along the medium, and changing the driving frequency determines which mode is excited.

Nodes and Antinodes

In the context of standing waves, nodes are the points along the medium that always remain at zero displacement due to destructive interference, while antinodes are the points that achieve maximum displacement because of constructive interference. The spatial arrangement and number of antinodes directly reflect the harmonic being excited.

Wave Speed and Tension Relationship

For waves on a string, the speed of the wave is determined by the properties of the string, particularly the tension and the linear mass density, as given by the formula v = ?(T/?). Changes in tension will affect the wave speed and consequently the resonant frequencies of the standing waves, thereby impacting the formation of normal modes.

Transcript

00:03 I know that the number of antinodes is currently three.

00:13 And so the number of nodes is the number of antinodes, anti -nodes, plus one.

00:22 So the number of nodes is currently four.

00:28 Frequency, no, no, that's not what i wanted to write.

00:36 M equals 2l over wavelength.

00:46 Now we are given f sub -0 and we're going to double it to 2 f -0.

00:58 The wave speed is the wavelength over frequency.

01:05 So i'm going to substitute in vf for wave.

01:12 Wavelength, m is 2l over vf.

01:24 And so, m is inversely proportional to f.

01:36 So the new one over the old number of nodes, which is four, is going to equal the new frequency, which is two, times the old frequency in the denominator and the old frequency in the numerator.

02:01 So new over this is old equals old over new.

02:11 Okay.

02:17 So all i have to do is solve this for m new.

02:19 M new is 4 over 2, which is 2.

02:28 So the number of nodes is two.

02:36 Notes.

02:39 So the number of antinodes is one less than that.

02:42 So there's going to be one anti -node...