Two Transverse Sinusoidal waves combine in a medium

[Willjeezy]

Homework Statement

Two Transverse Sinusoidal waves combine in a medium are described by the wave functions:

y1 = 3sin∏(x + 0.600t)

y2 = 3sin∏(x - 0.600t)

what is y1 + y2?

Homework Equations

the hint is that I am supposed to use:
sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

The Attempt at a Solution

the answer in the back is:
y = y1 + y2 = 6sin(∏x)cos(0.600∏t)

I am not really sure how they got that. In fact,the notation of y1 and y2 confuse me because y1 says:

y1=3sin∏(x + 0.600t)

and I interpret this as 3 multiplied by sin∏ multiplied by (x+0.600t)

but the problem is sin∏ = 0, so don't both equations just reduce to 0?

y1 = 0
y2 = 0?

Can someone clarify this for me?

 

[rude man]

y1=3sin∏(x + 0.600t)

and I interpret this as 3 multiplied by sin∏ multiplied by (x+0.600t)

No. 3sin[π(x + 0.6t)] means "three times the sine of the angle π(x + 0.6t)".

 

[Willjeezy]

hmm.3sin[π(x + 0.6t)]

it was written verbatim as:
3sin∏(x + 0.600t)

thanks, rude man.

 

[rude man]

hmm.

I wasn't sure if it was a typo, they are missing the outer square brackets

3sin[π(x + 0.6t)]

it was written verbatim as:
3sin∏(x + 0.600t)

thanks, rude man.

I just added the square brackets for emphasis for your benefit.

Why would anyone write "sinπ" when that quantity is zero?

 

[Nickie]

I can provide a response to this content by explaining the concept of superposition and how it applies to this problem.

Superposition is the principle that states when two or more waves meet at a point in a medium, the resulting displacement at that point is equal to the sum of the individual displacements of each wave. In other words, when two waves combine, they do not cancel each other out, but rather their amplitudes add together.

In the given problem, we have two transverse sinusoidal waves, y1 and y2, that are traveling in opposite directions along the same medium. The wave functions for these waves are given by y1 = 3sin∏(x + 0.600t) and y2 = 3sin∏(x - 0.600t). As the hint suggests, we can use the trigonometric identity sin(α + β) = sin(α)cos(β) + cos(α)sin(β) to simplify the expression for y1 + y2.

Using this identity, we can rewrite y1 and y2 as follows:

y1 = 3sin∏x cos(0.600t) + 3cos∏x sin(0.600t)
y2 = 3sin∏x cos(0.600t) - 3cos∏x sin(0.600t)

Now, when we add y1 and y2, the terms with cos(0.600t) will cancel each other out, leaving us with:

y1 + y2 = 6sin∏x cos(0.600t)

This is the same expression given in the back of the book. So, in conclusion, y1 + y2 does not reduce to 0, but rather it simplifies to 6sin∏x cos(0.600t). This shows that the combined wave has a larger amplitude than the individual waves, demonstrating the principle of superposition.