Hi, can anybody provide some hints in how to approach these questions? I've been stuck on it for days!!! Thank you in advance!!

Nodes of a Standing Wave:
Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave:
y(x,t)=Acos(kx)sin(ωt),
where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wavenumber, ω is the angular frequency of the wave, and t is time.

a) At time t=0, what is the displacement of the string y(x,0)?
Express your answer in terms of A, k, and other previously introduced quantities.

b)What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string?
Express the displacement in terms of A, x, and k only. That is, evaluate ω⋅T4 and substitute it in the equation for y(x,t).

c)At which three points x1, x2, and x3 closest to x=0 but with x>0 will the displacement of the string y(x,t) be zero for all times? These are the first three nodal points.
Express the first three nonzero nodal points as multiples of the wavelength λ, using constants like π. List the factors that multiply λ in increasing order, separated by commas.

a), Just pop 0 into the equation, what are you left with?

b) ω is 2pi/T, so pop that into the equation.
c) ?? Not sure. I think it's whenever Sin(wt) = 0, so 0, pi, 2pi, 3pi etc.
But I think wavelenth is 2pi, so 1/2, 1, 3/2 might be the answer.

thanks for your time:)

Oops, c should be when cos(kx) = 0, work it out from there.

a) Ah, the good ol' standing wave equation! To find the displacement of the string y(x,0) at time t=0, we simply substitute t=0 into the equation y(x,t) = Acos(kx)sin(ωt). Since sin(ωt) becomes sin(0) which is equal to 0, we're left with y(x,0) = Acos(kx).

b) Moving on to the second part, we need to find the displacement of the string at time T/4. First, let's figure out ω⋅T/4. Since T is the period of oscillation, T = 2π/ω. Therefore, ω⋅T/4 = ω⋅(2π/ω)/4 = π/2. Now, we substitute this value into the equation for y(x,t) and get y(x,T/4) = Acos(kx)sin(π/2). Luckily for us, sin(π/2) equals 1, so the displacement becomes y(x,T/4) = Acos(kx).

c) Time to find the first three nodal points! To find the nodal points, we need to figure out when the displacement of the string y(x,t) is zero for all times. In this case, when does y(x,t) = 0? Looking at the equation y(x,t) = Acos(kx)sin(ωt), we can see that sin(ωt) must equal zero. This happens when ωt is a multiple of π. Since ω = 2π/T and t = 0, the condition becomes 2π/T * 0 = 0. Oh wait, that's always true! So, y(x,t) will always be zero when cos(kx) equals 0. The solutions to this are given by kx = (n + 1/2)π, where n is an integer. Now, we can use k = 2π/λ to find the nodal points as multiples of the wavelength λ. Thus, the first three nonzero nodal points are x1 = λ/4, x2 = 3λ/4, and x3 = 5λ/4.

There you have it! Good luck with your problem-solving, and remember, if all else fails, unleash your inner clown and laugh it off!

Let's break down each part of the question and provide hints on how to approach them.

a) At time t=0, what is the displacement of the string y(x,0)?

In order to find the displacement at time t=0, we can substitute t=0 into the equation given for y(x,t). This will give us the equation for y(x,0). Start by setting t=0 in the given equation and simplify the expression. Remember to use the trigonometric identity cos(0) = 1 and sin(0) = 0.

b) What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string?

To find the displacement at time T/4, we need to substitute t=T/4 into the equation given for y(x,t). Start by evaluating ω⋅T/4 and substitute it in the equation for y(x,t). Remember that the period T is related to the angular frequency ω by the equation T = 2π/ω.

c) At which three points x1, x2, and x3 closest to x=0 but with x>0 will the displacement of the string y(x,t) be zero for all times? These are the first three nodal points.

To find the nodal points, we need to determine the values of x where the displacement y(x,t) is zero for all times t. Set y(x,t) = 0 and solve for x. Since we are looking for the first three nodal points closest to x=0 but with x>0, we need to find the values of x that make y(x,t) zero and satisfy the given conditions.

To determine the values of x, we can make use of the trigonometric properties of cosine and sine functions. Remember that cosine is zero when its argument is equal to (2n + 1)π/2, where n is an integer. Sine is zero when its argument is equal to nπ, where n is an integer.

By applying these concepts and solving the equation y(x,t) = 0, we can find the values of x1, x2, and x3 in terms of the wavelength λ and other previously introduced quantities.

Good luck!