The following wave function represents a travelling wave:
y = 3.45 / [6.35 + (6.35x−8.30 t )2]
where y and x are in cm, and t is in seconds.a) What is the maximum displacement of the wave at t = 7.50 s ?
b) what is the velocity.
Do you mean
y = 3.45 / [6.35 + (6.35x−8.30 t )^2] ??
if so for part a we want the shape when t = 7.5
y = 3.45/[ 6.35 + (6.35x - 62.25)^2]
that is biggest when 6.35x =62.25
for part b, when is 6.35 x - 8.30 t a constant? That is where if we hold that constant the wave looks the same.
y = f(6.35x-8.30t)
constant when
6.35x-8.30t = k
x = (8.3/6.35)t + k/6.35
dx/dt = 8.3/6.35
The following wave function represents a travelling wave:
y = 3.45 / [6.35 + (6.35x−8.30 t )2]
where y and x are in cm, and t is in seconds. a). What is the maximum displacement of the wave at t = 7.50 s ?
b). what is the wave velocity?
please explain more
If you move along with the wave, so that the wave always looks the same beside you, that is the phase velocity of the wave.
wave is a function of form:
y = f(ax-bt)
as long as ax-bt is constant, the function is constant
that happens when
a x - bt = constant
or
a dx/dt = b
dx/dt = b/a
In part a, the function is big when the denominator is small.
That is when
6.35x−8.30 t = 0
To find the maximum displacement of the wave at t = 7.50 s, we need to evaluate the given wave function at t = 7.50 s.
a) To do this, substitute t = 7.50 s into the wave function:
y = 3.45 / [6.35 + (6.35x−8.30 * 7.50 )^2]
Simplifying further,
y = 3.45 / [6.35 + (6.35x−62.25)^2]
Now, we need to find the maximum value of this function. To do that, we can differentiate the function with respect to x and set the derivative equal to zero:
dy/dx = 0
Differentiating the function:
dy/dx = (3.45 * 2 * (6.35x−62.25) * 6.35) / [6.35 + (6.35x−62.25)^2]^2
Setting dy/dx equal to zero and solving for x:
(3.45 * 2 * (6.35x−62.25) * 6.35) / [6.35 + (6.35x−62.25)^2]^2 = 0
Simplifying further,
(6.35x−62.25) / [6.35 + (6.35x−62.25)^2] = 0
6.35x−62.25 = 0
6.35x = 62.25
x = 62.25 / 6.35
x ≈ 9.80 cm
Now that we know the value of x, we can substitute it back into the wave function to find the maximum displacement at t = 7.50 s:
y = 3.45 / [6.35 + (6.35 * 9.80 − 8.30 * 7.50)^2]
Solving this equation will give us the maximum displacement of the wave at t = 7.50 s.
b) To calculate the velocity of the wave, we need to find the rate of change of displacement with respect to time:
v = dy/dt
Differentiating the wave function with respect to t:
v = -3.45 * (-8.30) / [6.35 + (6.35 * 9.80 − 8.30 * 7.50)^2]^2
Evaluating this equation will give us the velocity of the wave.