A standing wave pattern is observed in a thin wire with a...

A standing wave pattern is observed in a thin wire with a length of 3.00 $\mathrm{m}$ . The wave function is $$ y=(0.002 \mathrm{m}) \sin (\pi x) \cos (100 \pi t) $$ where $x$ is in meters and $t$ is in seconds. (a) How many loops does this pattern exhibit? (b) What is the fundamental frequency of vibration of the wire? (c) What If? If the original frequency is held constant and the tension in the wire is increased by a factor of $9,$ how many loops are present in the new pattern?

A standing wave pattern is observed in a thin wire with a length of 3.00 $\mathrm{m}$ . The wave function is
$$
y=(0.002 \mathrm{m}) \sin (\pi x) \cos (100 \pi t)
$$
where $x$ is in meters and $t$ is in seconds. (a) How many loops does this pattern exhibit? (b) What is the fundamental frequency of vibration of the wire? (c) What If? If the original frequency is held constant and the tension in the wire is increased by a factor of $9,$ how many loops are present in the new pattern?

Key Concepts

Standing Waves

Standing waves are produced by the superposition of two waves traveling in opposite directions. In systems with fixed boundaries, such as a vibrating wire or string, this interference leads to a pattern of nodes (points of zero amplitude) and antinodes (points of maximum amplitude) that do not move, creating a stable vibration pattern.

Nodes and Loops

Nodes are points where the medium remains at rest, while the sections between consecutive nodes form loops (or antinodes) that visibly bulge during oscillation. In a standing wave on a string fixed at both ends, the number of loops corresponds to the number of half-wavelength segments that fit into the length of the string.

Harmonic Modes

Harmonic modes are the specific patterns of vibration that occur on a string fixed at both ends, where only certain wavelengths are allowed. The n-th harmonic has the form sin(n?x/L), and the number of loops (or antinodes) in the standing wave equals n, with the arrangement determined by the boundary conditions.

Fundamental Frequency

The fundamental frequency is the lowest resonant frequency at which a string fixed at both ends will vibrate. It is determined by the wave speed and the length of the string according to the relation f? = v/(2L). Higher harmonics have frequencies that are integer multiples of the fundamental frequency.

Wave Speed and Tension

The speed of a wave on a string is given by v = ?(T/?), where T is the tension and ? is the linear mass density. Increasing the tension in the string increases the wave speed, which in turn affects the resonant conditions (i.e., the wavelengths and corresponding mode numbers) for a given driving frequency.

Transcript

00:01 For this problem on the topic of superposition and standing waves, we are given the wave function for a standing wave pattern observed net than wire.

00:09 We are asked to then calculate the loops this pattern will exhibit the fundamental frequency of the vibration of the wire as well as the number of loops that will be present if the tension in the wire is increased by a factor of 9.

00:23 So we are given the wave function, y is equal to 0 .002, sine pi x times the cosine of 100 pi t.

00:33 Now we know that the first term is twice the amplitude.

00:40 The term in the brackets for sine, which is pi x, we know is kx, and in the brackets for the cosine is omega -t.

00:58 So from here we can see that the wave number k, which we know is 2 pi over lambda, can be read off to be simply pi per meter.

01:12 And so we can find the wavelength lambda to be two meters, as well as the angular frequency omega, which we know is 2 pi f to be 100 times pi radiance per second.

01:31 And so we can find the linear frequency f simply to be 50 hertz.

01:39 Now with all of this information, we can first calculate the distance between adjacent nodes.

01:48 So the distance between adjacent nodes, we'll call it d and n, is half the wavelength.

01:53 So lambda over 2.

01:56 So this is simply one meter, since we know the wavelength lambda is 2 meters.

02:02 And hence we can calculate the number of loops on the string.

02:05 The number of loops on the string is the total length of wire l divided by the distance between the successive nodes, d and n.

02:16 And this is the total length of wire, 3 meters, divided by the distance between successive nodes 1 meter, and hence we see that there are 3 loops on this length of wire.

02:31 And that's our first answer...

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