The max of 2 cos: int[0,L] A*sin(kx-wt) dx

[Joseph123]

Hello,
Im stuck so for my final project i have to do some calculations:
$$∫_0^L〖A*sin⁡(kx-ωt)dx〗$$ This is an integral for a expresion of a sound wave trough a membrane
I have to get A out of the expression
$$U(t)=W* A* (-(cos⁡(kL-ωt)-cos⁡(-ωt))/k)$$So it got this i think till here everything is correct but now i have to get A(amplitude) for the max displacement this is when: $$cos⁡(ωt)-cos⁡(kL-ωt)$$ is at its maximum value out of logic i think its 2 because cos is between 1 and -1 so if the one is 1 and the other is -1 its 2 but i have no idea
how to proof this i tried it with the simpson formulas but i think im soo wrong: $$-2 sin⁡(((kL-ωt)-ωt)/2)*sin⁡(((kL-ωt)+ωt)/2) =-2 sin⁡(ωt-kL/2)*sin⁡(kL/2)$$$$ωt-kL/2=(-π)/2*n+k*2π (kϵz)$$$$kL/2=π/2-ωt*n+k*2π (kϵz)$$and this is then the answer:
$$U_max=W*A/k*2$$could just someone help me with getting the max value of those 2 cos it would help me soo much!

 

[blamocur]

Use trigonometric identity for $\cos\alpha - \cos\beta$ ?

 

[Cubist]

I haven't checked the first part of your work. After this line...

$$-2 sin⁡(((kL-ωt)-ωt)/2)*sin⁡(((kL-ωt)+ωt)/2) =-2 sin⁡(ωt-kL/2)*sin⁡(kL/2)$$

...I think you went a bit wrong. The variable is t isn't it? Therefore you effectively have...
$$X(t) = -2 \sin⁡(f(t))\sin⁡(kL/2)$$The kL/2 is constant isn't it? And f(t) is some function of the variable t. The maximum and minimum values of the above expression should be obvious at this point if you assume f(t) could take any value

Don't forget $U(t) = W*A*X(t)/k$

 

[Cubist]

Hello,
Im stuck so for my final project i have to do some calculations:
$$∫_0^L〖A*sin⁡(kx-ωt)dx〗$$ This is an integral for a expresion of a sound wave trough a membrane
I have to get A out of the expression
$$U(t)=W* A* (-(cos⁡(kL-ωt)-cos⁡(-ωt))/k)$$So it got this i think till here everything is correct but now i have to get A(amplitude) for the max displacement this is when: $$cos⁡(ωt)-cos⁡(kL-ωt)$$ is at its maximum value out of logic i think its 2 because cos is between 1 and -1 so if the one is 1 and the other is -1 its 2 but i have no idea
how to proof this i tried it with the simpson formulas but i think im soo wrong: $$-2 sin⁡(((kL-ωt)-ωt)/2)*sin⁡(((kL-ωt)+ωt)/2) =-2 sin⁡(ωt-kL/2)*sin⁡(kL/2)$$$$ωt-kL/2=(-π)/2*n+k*2π (kϵz)$$$$kL/2=π/2-ωt*n+k*2π (kϵz)$$and this is then the answer:
$$U_max=W*A/k*2$$could just someone help me with getting the max value of those 2 cos it would help me soo much!

In the first part, are you saying that...
$$\int_0^LA\sin⁡(kx-ωt)dx$$$$=-WA[\cos⁡(kL-ωt)-\cos⁡(-ωt)]/k$$
I think this would be true if W = 1 . (Also this integral assumes everything else is constant while x changes)

 

[Joseph123]

I haven't checked the first part of your work. After this line...

...I think you went a bit wrong. The variable is t isn't it? Therefore you effectively have...
$$X(t) = -2 \sin⁡(f(t))\sin⁡(kL/2)$$The kL/2 is constant isn't it? And f(t) is some function of the variable t. The maximum and minimum values of the above expression should be obvious at this point if you assume f(t) could take any value

Don't forget $U(t) = W*A*X(t)/k$

Thank you so much i think i understand it a little this is way above my league but you say that f(t) = $$(kL−ωt)−ωt)/2$$ is that correct?

 

[Cubist]

Thank you so much i think i understand it a little this is way above my league but you say that f(t) = $$(kL−ωt)−ωt)/2$$ is that correct?

I wrote f(t) because it doesn't really matter what f(t) actually is... it only matters that the value of f(t) changes when t changes. (I assume that you aren't given any constraints on the values that t can have).

Let's look at each factor of X(t), below, in order to work out the min and max values of X(t) when t changes

$$X(t) = -2 \sin⁡(f(t))\sin⁡(kL/2)$$
A) The first factor is -2. The value of this is a constant. It doesn't change when t changes.
B) The second factor is sin(f(t)). What is the maximum value that this could be when t changes? What is the minimum value that this could be when t changes?
C) The third factor is sin(kL/2). What happens to this value when t changes? The answer is nothing... because it doesn't contain "t". Therefore this is a constant value (that lies somewhere between -1 and 1 depending on the constants k and L).

Can you put this together to work out the min and max values of X(t)?

--

If you're still struggling, would you be able to work out the maximum value of this alternative expression:-
$$Z(t) = -2\sin(t)\sin(0.1)$$