A standing wave pattern on a string is described by y(x, t)=0.040(sin5 πx)(cos40 πt) where x and

step 1 16 .56. So we've got a standing wave on a string here. The SI units are assumed. So Y, this is i

A standing wave pattern on a string is described by y(x,...

A standing wave pattern on a string is described by $$y(x, t)=0.040(\sin 5 \pi x)(\cos 40 \pi t)$$ where $x$ and $y$ are in meters and $t$ is in seconds. For $x \geq 0,$ what is the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of $x ?$ (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For $t \geq 0$ what are the (g) first, (h) second, and (i) third time that all points on the string have zero trans- verse velocity?

A standing wave pattern on a string is described by
$$y(x, t)=0.040(\sin 5 \pi x)(\cos 40 \pi t)$$ where $x$ and $y$ are in meters and $t$ is in seconds. For $x \geq 0,$ what is
the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of $x ?$ (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e)
speed and (f) amplitude of the two traveling waves that interfere
to produce this wave? For $t \geq 0$ what are the (g) first, (h) second,
and (i) third time that all points on the string have zero trans-
verse velocity?

Key Concepts

Nodes

Nodes are specific points along the medium where the amplitude of the oscillations is always zero. They occur at positions where the interfering waves cancel each other completely, and their locations can be determined from the spatial part of the standing wave equation, typically involving sine or cosine functions that vanish at regular intervals.

Amplitude

The amplitude of a wave is the maximum displacement of points on the wave from their equilibrium positions. In standing waves, the amplitude varies spatially, reaching zero at the nodes and maximum values at the antinodes.

Transverse Velocity and Zero Crossings

Transverse velocity refers to the rate of change of displacement with respect to time at a given point on the medium. In a standing wave, there are specific times when the transverse velocity is zero at every point on the medium, corresponding to the instants when all points reach their maximum or minimum displacements.

Wave Speed

Wave speed is the rate at which the wave propagates through the medium. It is determined by both the angular frequency and the wave number (which is related to the spatial variation of the wave), and is given by the relation v = ?/k, where ? is the angular frequency and k is the wave number.

Standing Waves

Standing waves are produced by the interference of two waves of identical frequency and amplitude traveling in opposite directions. This interference results in fixed points of zero amplitude (nodes) and points of maximum amplitude (antinodes) that do not move in space, even though each point on the medium oscillates in time.

Superposition Principle

The superposition principle states that when two or more waves meet, the resultant displacement at any point is the algebraic sum of the displacements due to each individual wave. This principle underlies the formation of standing waves from two oppositely moving traveling waves.

Traveling Waves

Traveling waves are solutions to the wave equation that move energy through a medium. They are characterized by a definite speed and amplitude, and when two traveling waves of equal amplitude, frequency, and opposite direction superpose, they form a standing wave.

Angular Frequency and Period

Angular frequency is a measure of how rapidly the oscillatory motion occurs in radians per second, and it is directly related to the period of the oscillation. The period of a wave is the time it takes to complete one full cycle of oscillation, and it can be found by inverting the frequency.

Transcript

00:01 16 .56.

00:03 So we've got a standing wave on a string here.

00:09 The si units are assumed.

00:11 So y, this is in meters, x is in meters.

00:15 This is 5 pi radiance for meter.

00:17 This is 40 pi radiance per second, but we're saving ourselves a little bit of bother with writing all that out.

00:24 And then the parentheses we need to separate the things with units from the symbols, et cetera, and so forth.

00:29 That you just, you know, it's a bit lazy, but sometimes laziness is okay.

00:36 So we want to find the location of the first three nodes, the ones with the smallest three x values for x graded n or equal to zero.

00:48 We want to find the period of the oscillatory motion of any point that isn't a node.

00:56 Obviously, since the nodes don't move, talking about them having a period of oscillation is a little bit.

01:01 Bit meaningless.

01:03 Then we have two traveling waves that interfere to produce the standing waves.

01:10 We want to find their speed and amplitude, and then we want to find the first three times after t equals zero where all points on the string have a zero transverse velocity.

01:33 So we have a number of things to do, but some of them are kind of you do them all at once.

01:41 So we want the nodes are going to be given by sign of 5 pi x equals 0.

01:57 This implies that 5 pi x is equal to 0, pi, 2 pi, etc., which means that x is going to be 0, 1 5th, 2 fifths, 3 5ths, and so on forever.

02:34 Or until you reach the end of the string.

02:41 So then the smallest ones, we can just add an m for meters on here and express them as decimals.

02:47 And so we have one at zero meters, one at 0 .2 meters, and then one at 0 .4 meters.

02:56 And then every 0 .2 meters after that will have another node, but we only care about the ones with the three smallest x values.

03:20 So every point except for the nodes has a simple harmonic motion, with the frequency that's just omega over 2 pi...

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