Harmonic wave / wave problem

[matywlee]

Homework Statement

Please kindly help me.
Actually I don't quite understand what the meaning of harmonic wave is and the mathematics that expresses it.
h(x,y;t) = h sin(wt-kx+δ)
h represents the position of the particle in a particular time? Or the wave motion?
What is the physical meaning of w, k? What are they describing? Why the function is written as h sin(wt-kx+d)?
Can you explain that? My mathematic is not very good, to be honest.

1) What is the distance between two maxima/minima in an interference pattern of two waves u1 and u2 described by
u1(x) = cos(k1 x); u2(x) = cos(k2 x)
What happens when k1=k2?

2) "The so-called nami-water is good to our body!" Is the statement scientific (falsifiable), and why?

Homework Equations

The Attempt at a Solution

The distance between two maxima/minima in an interference pattern = the interfered wave's wavelength?

 

[haruspex]

h(x,y;t) = h sin(wt-kx+δ)
h represents the position of the particle in a particular time? Or the wave motion?

That equation doesn't make much sense. You have h both sides (do you mean h(x,y;t) = A sin(wt-kx+δ), or maybe h(x,y;t) = h_max sin(wt-kx+δ)?), and y does not appear on the right.
If you mean h(x, t) = A sin(wt-kx+δ), that is defining a function.
It helps to understand that when we write y=y(x) we make a 'pun'. The y on the left is a variable; the y on the right is a function. They are not really the same thing. But the practice is so standard that e.g. y(x) = 2x is commonly used as a shorthand for y = y(x) = 2x. I.e. defining the function y(x) is taken as an implicit definition of a variable of the same name.
In the present case, the equation h(x, t) = A sin(wt-kx+δ) defines a function h(x, t) and a variable h. The variable h represents a displacement from an average state (position, usually). So the answer to your question is 'both'.

What is the physical meaning of w, k? What are they describing? Why the function is written as h sin(wt-kx+d)?

If we fix some point along the line x, we get h = A sin(wt+c). This shows that h varies over time, repeating every interval 2π/w: sin(w(t+2π/w)+c) = sin(wt+2π+c) = sin(wt+c). So the frequency is w.
If we fix on a point in time and look along the line, we see a shape that repeats every 2π/k. So we say the wavelength is 2π/k.
If we fix on some peak in the curve and ask how that moves over time, we want wt-kx = constant. I.e. x = (w/k)t + constant. This means that the wave pattern moves at speed w/k.

1) What is the distance between two maxima/minima in an interference pattern of two waves u1 and u2 described by
u1(x) = cos(k1 x); u2(x) = cos(k2 x)

So what is the equation for the combined wave?
Do you know any trig formula that allows you to write that differently?

 

[matywlee]

Yes, I mean h(x, t) = A sin(wt-kx+δ).

1) What is the distance between two maxima/minima in an interference pattern of two waves u1 and u2 described by
u1(x) = cos(k1 x); u2(x) = cos(k2 x)
So what is the equation for the combined wave?
I don't know. It is just shown for me this question. So I ask it here. Do you have any idea?

 

[haruspex]

u1(x) = cos(k1 x); u2(x) = cos(k2 x)
So what is the equation for the combined wave?

Most obviously, it's u(x) = u1(x)+u2(x) = cos(k1 x)+ cos(k2 x). But to answer the question it will help to write this differently. Do you know a trig formula involving cos(A)+cos(B)?