A transverse wave on a string is described by the equation y(x, t)=(0.350 m) sin[(1.25 rad / m)

step 1 Let's go ahead and copy down that wave function. Y of XT is equal to 0 .35 meters times the sign

A transverse wave on a string is described by the equation...

Key Concepts

Temporal Analysis and Time Interval Calculation

Temporal analysis in the context of waves involves examining the periodic behavior of the waveform to determine specific instants when certain displacements occur. By considering the periodicity and phase of the wave, one can calculate the time intervals between instances when the wave reaches particular positions, which is crucial for understanding wave dynamics and synchronization phenomena.

Phase and Phase Shift

The phase of a wave includes the combined effects of spatial and temporal variations, indicating the state of the oscillation at any point along the wave. A phase shift occurs when the argument of the sine or cosine function is altered, leading to a displacement in time or space. Understanding phase shifts is essential for analyzing when a waveform reaches specific values during its oscillatory cycle.

Wave Speed

Wave speed is the rate at which the wave propagates through the medium and can be calculated by relating the angular frequency and the wave number. This concept ties together the spatial and temporal characteristics of the wave, providing insight into how fast disturbances travel along the medium.

Sinusoidal Wave Equation

A sinusoidal wave equation, typically involving a sine (or cosine) function, represents periodic oscillations in both space and time. This mathematical form encapsulates the wave’s amplitude, spatial variation, and temporal evolution, allowing for the analysis of wave properties and behavior across different positions and instants.

Transverse Waves

Transverse waves are disturbances that move through a medium in which the displacement of the particles is perpendicular to the direction of wave propagation. This type of wave is characterized by oscillatory motion at right angles to the direction in which the wave travels, making it a fundamental concept in understanding phenomena like vibrations in strings and electromagnetic waves.

Wave Parameters (Amplitude, Wave Number, Angular Frequency)

The key parameters in a wave’s equation include the amplitude, wave number, and angular frequency. The amplitude indicates the maximum displacement from equilibrium, the wave number determines the spatial frequency (how many cycles occur per unit distance), and the angular frequency measures how rapidly the wave oscillates in time. Together, these parameters define the wave’s energy, periodicity, and propagation characteristics.

Transcript

00:01 Let's go ahead and copy down that wave function.

00:03 Y of xt is equal to 0 .35 meters times the sign of 1 .25 radiance per meter times x minus, excuse me, plus 99 .6 radiance per second times t.

01:03 Great.

01:05 So now we're going to consider the element of the string at x equals 0.

01:10 And we want to know what is the time interval between the first two instance when this element has a position of...

01:17 So we're going to just substitute in x equals 0 and y equals 0 .175 meters.

01:37 0 .175 meters, right? we're going to use that equation right there to solve for t.

01:45 And now because this is a sign function, we should have actually infinitely many...

01:52 Points in time that this occurs, right? so let's just go ahead to fill those numbers in.

02:17 I'm going to ignore units just for now just to save some space.

02:24 This is being multiplied by zero, so i won't even write that in there.

02:30 99 .6.

02:34 99 .6 times t, right? yeah, so i mean, let's just divide by 0 .35 .0 .0.

02:58 0 .175 by 0 .35 .05.

03:06 0 .5 is sign of 99 .6 t.

03:16 We can take the arc sign of this and see what we get.

03:26 So 99 .6t is equal to the arc sign of 0 .5.

03:33 Yeah, so i mean the arc sign of 0 .5, that's like at which angles is the sign equals to 0 .5? well, at this occurs at, i think it's pi over 6.

03:58 Yeah.

04:00 So i think we got pi over 6.

04:01 So 99 .6t is pi over 6.

04:07 And it also occurs on the other side of the unit circle, which, let's see, so we got pi over six, then pi over four, then pi over three, and then pi, or no, yeah, pi, pi for two.

04:31 And then we got, i think it's, i think it should be, so that's three pi over six, right? so then we got four pi over six, right? and that not there, but keep going.

04:45 4 pi over 6.

04:46 Then we got, i think it should be 2 pi over 3, right? or maybe it's 5 pi over 6, 5 pi over 6.

05:04 5 pi over 6, i think.

05:12 Let me use a calculator just to be sure.

05:14 Because i'm doing the unit circle in my head right now.

05:17 Yeah, those two are the correct angles.

05:20 So now we will just take our, now we'll just solve for the t's.

05:24 The two time intervals here.

05:27 Pi over six, that's going to be over 99 .6 times six.

05:35 And this one will be 5 pi over 99 .6 times six...