Plane Wave Propogation (help understanding steps in book)

[FrogPad]

Plane Wave Propagation (help understanding steps in book)

Basically this problem requires that $\vec H$ vanishes at $t = 3ms$. I don't understand how they did the last step, here is the solution:

$$\vec H = \hat z 4 \times 10^{-6} \cos \left(10^7 \pi t - k_0 y +\frac{\pi}{4} \right)$$

$$k_0 = \omega \sqrt{\mu_0 \epsilon_0} = \frac{10^7 \pi}{3 \times 10^8} = 0.105 \,\,\,\, (rad/m)$$
$$\lambda = \frac{2\pi}{k_0} = 60 \,\,\,\, (m)$$

At $t = 3 \times 10^{-3} \,\,\,\, (s)$ we require the argument of cosine in $\vec H$:

$$10^7 \pi (3 \times 10^{-3}) - \frac{\pi}{30}y +\frac{\pi}{4} = \pm n \pi + \frac{\pi}{2}, \,\,\,\,n=0,1,2,\ldots$$

$$y = \pm 30n - 7.5 \,\,\,\, (m)$$

Now how the hell do they get the last line? The... $y = \pm 30n - 7.5 \,\,\,\, (m)$ (Lets call this (1)) line.
They drop $10^7 \pi (3 \times 10^{-3})$ why?

My problem is that if we plug in (1) into,
$$\vec H = \hat z 4 \times 10^{-6} \cos \left(10^7 \pi t - k_0 y +\frac{\pi}{4} \right)$$

and set t = 0.003, then if we feed the function values of n then the cosine doesn't drop to 0.

Thanks in advance!

 

[OlderDan]

The term they dropped is a multiple of pi (and in fact a multiple of 2pi). Adding a multiple of pi to an angle at which the cosine is zero is still going to give you a cosine of zero.

 

[FrogPad]

The term they dropped is a multiple of pi (and in fact a multiple of 2pi). Adding a multiple of pi to an angle at which the cosine is zero is still going to give you a cosine of zero.

Ahh... tricky tricky. (I mean, yeah, it's not too tricky but still, I guess I will add this to my list of things to look for subconsciously while evaluating expressions)