What does it mean that all points between two adjacent nodes in a standing wave oscillate in phase? I sort of get what in phase means, it means that the peaks and troughs etc of 2 waves align. But how can we say that a single standing wave is in phase? Can someone please explain to me what it means that "that all points between two adjacent nodes in a standing wave oscillate in phase?" And please, try to do this at the level of a high school sophomore who still hasn't learnt Calculus based physics. Thank you.
It means that all those points go up at the same time and down at the same time.
The do not go equally high up and down. Their amplitudes are different. But they do it at the same time nevertheless.
For general waves, in-phase means that the points of two waves progress (move) equally. They "follow each other" perfectly. In standing waves, this boils down to the points not progressing, but still "following each other" by rising and falling equally.
Consider a sine function whose amplitude is time dependent. It can be written as $y(x,t)=A(t)sin(x)$ where $t$ is time, $x$ is the abscissa($x$-coordinate) and $y$ is the ordinate($y$-coordinate) which in this case is also time dependent due to the time dependence of the amplitude of the sine wave. Now choose an abscissa on the wave at a time $t_1$, say $x_1$ where the value of the function will be $y(x_1,t_1)$. Because the sine wave has a periodicity of $2\pi$ you will find an abscissa $x_2$ which will give you the same value of ordinate as given by $x_1$. Mathematically, $y(x_1,t_1) = y(x_2,t_1)$ but these abscissa's will be related by $x_2 = x_1 \pm 2n\pi$. The different abscissa which give same value of ordinate at a given time are said to be in phase. So $x_1$ and $x_2$ will be in phase. Now the role of time is that if you vary time then the two abscissa which were in phase will always remain in phase as the ordinate varies, i.e. $y(x_1,t_1) = y(x_2,t_1)$ and $y(x_1,t_2) = y(x_2,t_2)$. Hope this helps!!
Use the PhET Wave on a String program with the settings as shown in the screenshot below.
Run the program and let it settle down for a few minutes when you should see that a standing wave has been formed.
Positions $B$ and $C$ are nodes.
Observe the red “particles” between $B$ and $C$ and this is where you will see the particles moving “in phase with one another” but with a differing amplitude.
The red “particles” between $A$ and $B$ will also be moving in phase with one another.
Also note that the motion between $AB$ is $180^\circ$ out of phase with that between $BC$ .
Using the “pause” button or the “Slow Motion” setting might help you visualise what is happening.