The discussion revolves around the maximum vibration speed of a traveling wave described by the equation y(x,t) = (9.00cm)sin((5Πcm-1)x + (3Π rad/s)t - Π/8). Participants are exploring the meaning of 'vibration speed' and how to determine it from the given wave function.
The conversation is ongoing, with participants sharing insights and clarifications about the relationship between the wave function and the maximum speed. Some guidance has been provided regarding the conditions under which maximum speed occurs, but no consensus has been reached on the final interpretation or calculation.
Participants are navigating the complexities of a multi-part problem, with specific attention to the definitions and implications of terms used in the context of wave mechanics. There is a recognition of the need for clarity on the relationship between the wave's parameters and the physical interpretation of speed.
You didn't answer my question. There's no point guessing at procedures until you know what it is you're trying to find.Jadon said:The only thing I could think of was taking the derivative of the original equation (not sure whether with respect to x or t) and then setting equal to zero and solving for v (probably with respect to x then) but I would still be stuck with another variable.
Yes.Jadon said:Ok then, I see what you mean. I believe it means the maximum speed of the element of string. So the speed as the element is at y = 0.
y is a function of two independent variable, x and t: y = y(x,t). What direction does an element move in? Algebraically, what expression represents the speed of an element at position x and time t?Jadon said:Ok so, to find this speed, I would take the derivative of the function as one would do when doing this for oscillations, correct? I just am not sure how to do this with variables x and t inside the cosine. Or am I going the wrong direction?
The elements are not moving in the x direction. The wave moves in the x direction, but we're not concerned here with the speed of the wave.Jadon said:dx/dt would.
Exactly.Jadon said:it moves in the y direction.
You could do that, but as you already said it will occur when y = 0. What does that tell you about the argument of the cosine function in your dy/dt formula?Jadon said:Wouldn't I have to set the second derivative equal to zero to find the maximum velocity?
Yes, but as I wrote, you can get that from y = 0, without taking derivatives.Jadon said:Well if you were to use the second derivative, the sin function would equal 0, so maybe the cosine function equals 1?
haruspex said:What does that tell you about the argument of the cosine function in your dy/dt formula?
You mean dy/dt, right?Jadon said:I don't understand how to obtain any information from this. From dy/dx = 27π(cos((5πcm-1)x + (3π rad/s)t - π/8)
No, it's simpler that that. If sin(theta) = 0, what can you say about cos(theta)?Jadon said:That would tell me that sin-1(0) = 5πcm-1x + (3π rad/s)t - π/8 .
This in turn would tell me that π/8 = 5πcm-1x + (3π rad/s)t and that 1/8 = 5cm-1x + (3rad/s)t. I'm just not sure what direction I should be going, what I am truly solving for.
Yes.Jadon said:cos(theta) would equal 1 then, so the maximum vibration speed would be equal to 27π cm/s or rad/s (not quite sure on the units anymore lol).
Not sure what you are suggesting as the answer. Radians are dimensionless. If an arc of a circle radius 1cm subtends an angle of 2 radians at the centre, the length of the arc is 2cm, not 2cm-radians.RUber said:Think about what the units are telling you. You really have 9cm ##\times 3\pi ## rads per second. If ##3\pi## rads per second means 1.5 full cycles per second, and a cycle is from +9cm to -9cm, what is the speed of the string?