A sinusoidal wave has the following wave function:y(x,t) (2.5 m) sin[(3.0 m-1) x- (24 s-1) t + T/2]What is the frequency of this wave? (a) 0.48 Hz, (b) 3.0 Hz, (c) ...

A sinusoidal wave is described by the wave function
$$
y=(0.25 \mathrm{m}) \sin (0.30 x-40 t)
$$
where $x$ and $y$ are in meters and $t$ is in seconds. Deter- mine for this wave the (a) amplitude, (b) angular frequency, (c) angular wave number, (d) wavelength, (e) wave speed, and (f) direction of motion.

We know that why equals a times with sign of K X minus omega T and were given the equation y equals 0.25 times the sign of 0.3 x minus 40. Therefore, we know that the amplitude is equal to 0.25 meters. Next, we know that the angular frequency omega is equal 2 40 readings per second. Next, we know that the weave number K is equal 2.3 radiance per meter. Next to find wavelength, we know that K equals two pi divided by we've length. Therefore, wavelength equals two pi divided by K which is equal to two pi divided by 20.3. This gives us a wavelength that is equal to 20.94 meters. Next to find velocity, we know that the velocity is equal to the angular frequency over the wave number. This gives us 40 divided by 400.3 and this gives us a wave speed of 133 divided by point of 133.33 meters per second. Next, find direction. We need to look at this sign right here. Since that is negative, we know that the wave is traveling in the positive X direction

The amplitude and all the rest will be given by the wave function here. So your 0.25 m it was but meters and sign of 0.3. We're going to assume that's radiance per meter on that. This number over here is radiance per second, we'll probably find out. Alright. The problem doesn't say specifically what these numbers, what kind of units? These numbers come in. But just that this is in meters, all right. But we can assume its radiance, Premier and radiance per second. So, um, Theano attitude is right here, right here. The amplitude is 0.25 meters. What's next? The angular frequency angular frequency is right here. 40 radiance for second. The angular wave number, the angular wave number, or sometimes just called the wave number is 0.3 radiance. Her meter, the wavelength. So this is actually equal to, um, two pi over the wavelength which would make the wavelength equals two pi over 0.3. That's about 21 21 m. Um, the wave speed speed is equal to the wavelength times to frequency. So the frequency is, um, related to the angular frequency by the fact that the angular frequency is two pi times to your frequency. So it's gonna be 40/2 pi, which is No, I'm just gonna leave it. I'm just gonna leave it as in terms of pie, Um, 20 over pie hurts. Um, that is the waves. So that is the frequency. So now the speed is gonna be two pi over 0.3 times 20 over pie pies air gone. This is 40 over 0.3, which is 133.33 meters per second. That is the way of speed. And finally, the direction of motion is in the positive direction. And we know that because of this so mhm.

Yeah in discussion we have given silence or I'll be that is why equal to Few .25 m. Right? The lab I sign 0.3 X minus 40 P. No we have to calculate the completely angular frequency angular wave number violent, we speak so for this we know there uh genuine equation of there is why it was true. Yes. Sign E x minus omega mm. No, here is actually the completed observation. Omega is the angular frequency and K. Is the angular the number. Um on comparing the able to equations we get complete you if it calls to 0.25 mega comes to part B. So here we get angular frequency omega it will 40 billion per second comes to R. C. Anguilla's the numbers It was 2013 jerry in was a major No comes to our D. Here we need to calculate the big land lambda. We know that Waveland lambda. It was true. I divided by K. Therefore The substitute 0.3 four K. And the formula supply lambda equal to two point divided by 0.3. So from above we get λ equals two 20.94 meta now comes to All right mm. So here we have to find out the way the speed which is equal to F marty plumber lambda. No, you can like F as omega project by Dubai. Therefore very straight as equal to Omega multiple are divided by two. Why? Now on Putting the values we get 20 194 multiplied by 40 divided by two. Why? So we get we've spirit as equals two 133 .3. Meet up for a second. No. Yeah. Mentioned that in which direction the wave is strapping. So the very is traveling alone. Well. So do you X. Axis.

So here they're asking us again for all the properties of this wave. Um, the wave functions for both of these traveling waves that make up the standing wave and then the harmonic if the harmonic can be identified or not. So let's first try right this equation Why of X T equals four point four four millimeters and those would be sign of X times thirty two point five radiance per meter and then sign of t seven hundred and fifty four radiance per second. So at this point, we can relate this to the general equation for a standing wave. So we can say why equals x minus t equals to a sign of k x sign of Omega team. So we know that this thirty two point five radiance per meter is K. And we know that this seven hundred fifty for ratings per second is Omega. So their first asking us for the amplitude. So amplitude is going to be ah equal to four point four four meters millimetres over, too. There's going to be two point two two millimeters and we'LL write this in meters So we'll say point zero zero two two two meters then they want the rave with the wavelength. So we know that the wave number is equal to two pi over the wavelength. So that means that the wave length is going to be equal to two pi over the way of number. Yeah, to pie divided by thirty two point five. And this is going to give us a wavelength of point one nine three meters. At this point, they want us to find also the frequency. We know that Omega is going to equal to two pi times the frequency so the frequency is simply equal to Omega over to pie. We can say omega again is seven hundred fifty four divided by two pi and this is going to be one hundred and twenty hertz. So we simply have all of that. And then they want us to find the propagation speed which we have. So this will be the propagation. Speed is equal to the wavelength times the frequency. We have both of these values. So it's going to be point one nine three meters times one hundred and twenty hertz, and this will give us a propagation speed of twenty three point two immediate per second. Now this will be our D. Now they're asking us e what can me draw the Kimmy find the way functions for both of these travel of both of these traveling waves. So traveling in the plus X direction the wave would be why suborn x t with equal a co sign of K X minus omega t. And then, if we were Tio plug in for our knowns, this would be equal to point zero zero two two two meters and then co sign next Ah times thirty two point five radiance per meter and then minus see seven hundred and fifty four radiance per second. And that will be the way function for the travelling wave that is traveling in the plus X direction. So we'LL do travelling and the in minus X direction and this will be y sub to X except tea, a co sign and then k x It'LL be plus omega T instead of minus omega T. And at this point again we can simply just plug in for unknowns. Co sign of X times thirty two point five radiance per meter minus T seven hundred and fifty four radiance per second and then death is asking us. Can we find the harmonic? While the harmonic cannot be dent identified because the length of the string is not specified, so length of string unknown means that the harmonic is unknown as well. And that's the end of the solution. Thank you for watching.