Consider the following standing wave, (all units are SI)
z=0.2sin(0.4y)cos(350t)
(a) what is the wavelength?
(b) The values of y where the displacement is always zero, are of the form y=a+bn, where n=0,±1,±2,3, etc.
What is b ?
(c) The values of t where the displacement at all values of y equals to zero are of the form : t=c+dn, where n=0,±1,±2,3, etc.
What is c ?
what is d ?
d and e ate the most difficult
a) 15.705 ,
b) 350
c) 55.714
d) 0
other parts still working on it
To determine the answers to the questions, let's break down the given standing wave equation:
z = 0.2sin(0.4y)cos(350t)
(a) Wavelength can be determined by examining the argument of the sine function. In this case, the argument is 0.4y. The wavelength (λ) is inversely proportional to the coefficient of y. Therefore, the wavelength is given by:
λ = 2π/k
where k is the coefficient of y.
In this case, k = 0.4. Substituting this value into the equation gives:
λ = 2π/0.4 = 5π
So the wavelength is 5π.
(b) To find the values of y where the displacement is always zero, we need to find the zeros of the sine function. In this case, the equation is:
0.2sin(0.4y)cos(350t) = 0
Since the cosine function is nonzero for any value of t, we can ignore it for this question. Setting the sine function to zero gives:
sin(0.4y) = 0
The sine function is zero at multiples of π, so we can write:
0.4y = nπ
where n is an integer. Rearranging the equation, we get:
y = (nπ)/(0.4) = (5/2)nπ
Comparing this with the given form y = a + bn, we can see that b = 5/2.
(c) To find the values of t where the displacement is zero for all values of y, we need to find the zeros of the cosine function. In this case, the equation is:
0.2sin(0.4y)cos(350t) = 0
Now, we ignore the sine function and focus on the cosine function. Setting the cosine function to zero gives:
cos(350t) = 0
The cosine function is zero at odd multiples of π/2. Setting 350t equal to (2n+1)π/2, where n is an integer, we get:
350t = (2n+1)π/2
Simplifying the equation, we find:
t = (2n+1)π/(700)
Comparing this with the given form t = c + dn, we can see that c = 0 and d = π/(700).