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)?

 

[Nickie]

Hello, thank you for reaching out for help on the topic of harmonic waves. I will do my best to explain the concept and mathematics behind them.

A harmonic wave is a type of periodic wave that has a sinusoidal shape. This means that the wave repeats itself at regular intervals. The mathematics that expresses it is the equation you provided:

h(x,y;t) = h sin(wt-kx+δ)

In this equation, h represents the amplitude of the wave, or the maximum displacement of the particles in the wave. The variable t represents time, while x represents the position along the wave. The values of w, k, and δ are constants that describe the properties of the wave.

The physical meaning of w and k can be better understood by looking at the wave equation in more detail. The term wt-kx represents the phase of the wave, which determines the position of the wave at a given time. The term w represents the angular frequency of the wave, which is related to the frequency of the wave by the equation f = w/2π. The variable k represents the wave number, which is related to the wavelength of the wave by the equation λ = 2π/k. Essentially, w and k describe the properties of the wave and how it changes over time and space.

The reason why the function is written as h sin(wt-kx+δ) is because the sine function is a mathematical representation of a harmonic wave. It allows us to visualize and understand the properties of the wave, such as its amplitude, frequency, and phase.

Now, let's move on to the two questions you provided.

1) The distance between two maxima/minima in an interference pattern is indeed related to the wavelength of the wave. In the case of two waves, u1 and u2, the distance between two maxima/minima will depend on the superposition of the two waves. If k1 and k2 are different, the distance between the maxima/minima will be determined by the difference between the two wavelengths. However, if k1 and k2 are the same, the waves will have the same wavelength and the maxima/minima will overlap, creating a larger amplitude.

2) The statement "The so-called nami-water is good to our body!" is not a scientific statement because it is not falsifiable. In order for a statement to be scientific, it must be able to be tested and potentially proven wrong. However