The nodes of a wave are the points where the displacement of the wave is always zero. For a stationary wave, these occur where the spatial part of the wave function is zero. In this case, the spatial part of the wave function is cos(πx). The cosine function is zero where its argument is an odd multiple of π/2. So, we have:

πx = (2n+1)π/2

Solving for x gives:

x = (2n+1)/2

The three smallest positive values of x that correspond to nodes are therefore x = 1/2, 3/2, 5/2 (for n = 0, 1, 2).

The antinodes of a wave are the points where the displacement of the wave is at a maximum. For a stationary wave, these occur where the spatial part of the wave function is at a maximum or minimum. The cosine function is at a maximum or minimum where its argument is a multiple of π. So, we have:

πx = mπ

Solving for x gives:

x = m

The three smallest positive values of x that correspond to antinodes are therefore x = 1, 2, 3 (for m = 1, 2, 3).

Question

The nodes of a wave are the points where the displacement of the wave is always zero. For a stationary wave, these occur where the spatial part of the wave function is zero. In this case, the spatial part of the wave function is cos(πx). The cosine function is zero where its argument is an odd multiple of π/2. So, we have:

πx = (2n+1)π/2

Solving for x gives:

x = (2n+1)/2

The three smallest positive values of x that correspond to nodes are therefore x = 1/2, 3/2, 5/2 (for n = 0, 1, 2).

The antinodes of a wave are the points where the displacement of the wave is at a maximum. For a stationary wave, these occur where the spatial part of the wave function is at a maximum or minimum. The cosine function is at a maximum or minimum where its argument is a multiple of π. So, we have:

πx = mπ

Solving for x gives:

x = m

The three smallest positive values of x that correspond to antinodes are therefore x = 1, 2, 3 (for m = 1, 2, 3).

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The nodes of a wave are the points where the displacement of the wave is always zero. For a stationary wave, these occur where the spatial part of the wave function is zero. In this case, the spatial part of the wave function is cos(πx). The cosine function is zero where its argument is an odd multiple of π/2. So, we have:

πx = (2n+1)π/2

Solving for x gives:

x = (2n+1)/2

The three smallest positive values of x that correspond to nodes are therefore x = 1/2, 3/2, 5/2 (for n = 0, 1, 2).

The antinodes of a wave are the points where the displacement of the wave is at a maximum. For a stationary wave, these occur where the spatial part of the wave function is at a maximum or minimum. The cosine function is at a maximum or minimum where its argument is a multiple of π. So, we have:

πx = mπ

Solving for x gives:

x = m

The three smallest positive values of x that correspond to antinodes are therefore x = 1, 2, 3 (for m = 1, 2, 3).

Question 5A stationary wave is represented by the following expression:𝑦 = 5⁡cos𝜋𝑥⁡sin𝜋𝑡where y and x in centimeters and t in seconds. Determine:(a) the three smallest value of x (x >0) that corresponds toi. nodesii. antinodesPlease work out part 2

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