Consider the following standing wave, (all units are SI)

z=0.2sin(0.4y)cos(350t)

(a) 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 a?

(b) 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 ?

plz help.....!

To find the values of y where the displacement is always zero, we need to set the equation equal to zero:

0.2sin(0.4y)cos(350t) = 0

Since the expression involves both sine and cosine functions, the values of y that satisfy this equation will be determined by both functions.

First, let's look at the sine function:

sin(0.4y) = 0

To find the values of y that make the sine function equal to zero, we need to determine when the argument of the sine function, 0.4y, is equal to an integer multiple of π. In other words:

0.4y = nπ, where n is an integer

Solving for y, we divide both sides of the equation by 0.4:

y = (nπ) / 0.4

So, the values of y that satisfy the sine function are of the form:

y = (nπ) / 0.4

Now let's consider the cosine function:

cos(350t) = 0

To find the values of t that make the cosine function equal to zero, we need to determine when the argument of the cosine function, 350t, is equal to an odd multiple of π/2. In other words:

350t = (2n+1)(π/2), where n is an integer

Solving for t, we divide both sides of the equation by 350:

t = [(2n+1)(π/2)] / 350

So, the values of t that satisfy the cosine function are of the form:

t = [(2n+1)(π/2)] / 350

Now, let's focus on finding the specific values of a, c, and d.

(a) To find the value of a, we will consider the equation we obtained for y:

y = (nπ) / 0.4

Since the expression of y is in the form y = a + bn, where n is an integer, we need to determine the values of a and b. In this case, b is equal to (nπ) / 0.4, which means that b = π / 0.4. So the equation can be rewritten as:

y = a + (π / 0.4)n

Comparing this with the given form (y = a + bn), we can equate the coefficients:

a = 0

Therefore, a = 0.

(b) To find the values of c and d, we will consider the equation we obtained for t:

t = [(2n+1)(π/2)] / 350

Again, the expression of t is in the form t = c + dn, where n is an integer. In this case, d is equal to [(2n+1)(π/2)] / 350, which means that d = (π/2) / 350. So the equation can be rewritten as:

t = c + [(π/2) / 350]n

Comparing this with the given form (t = c + dn), we can equate the coefficients:

c = 0
d = (π/2) / 350

Therefore, c = 0 and d = (π/2) / 350.