Harmonic wave / wave problem

1. Jul 8, 2013

matywlee

1. The problem statement, all variables and given/known data
Actually I dont 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?

2. Relevant equations

3. The attempt at a solution
The distance between two maxima/minima in an interference pattern = the interfered wave's wavelength?

2. Jul 8, 2013

haruspex

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

3. Jul 12, 2013

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 dont know. It is just shown for me this question. So I ask it here. Do you have any idea?

4. Jul 12, 2013

haruspex

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)?