Animated Solution for Physics - Waves: A wave $y=a\sin (\omega t-kx)$ on a string meets with another wave producing a node at $x=0$. Then, the equation of the unknown wave is

Imagine you are holding one end of a long string and you send a wave pulse down its length. The problem tells us that this wave, let's call it $y_1$, meets another mysterious wave. When they collide and superimpose, they create a very specific pattern: a standing wave with a node exactly at the origin, where $x=0$.

Now, what exactly is a node? A node is a point of absolute stillness. It's a place where the string never moves up or down, no matter what time it is. This single piece of information is the master key to unlocking the entire problem.

So, how do we mathematically guarantee that a node forms at the origin? This is where the powerful principle of superposition comes into play. Superposition simply states that when two waves meet, their individual displacements add up algebraically to give the net displacement.

Therefore, if we want a node at $x=0$, the net displacement, $y_{net}$, must be exactly zero at that specific location. And this must hold true for every single instant of time $t$. It's not just zero for a moment; it's permanently zero. This gives us our master equation:

Let's take our given wave equation, $y_1=a\sin (\omega t-kx)$, and see how it behaves right at the origin. We simply substitute $x=0$ into the equation. What happens? The $kx$ term vanishes completely. We are left with:

This tells us that if the first wave were acting alone, the point at the origin would be oscillating up and down with an amplitude $a$ and angular frequency $\omega$. But we know the origin is a node, so it can't be moving! To keep the origin perfectly still, the second wave must perfectly cancel out the motion of the first wave. It's like a tug-of-war where both sides pull with exactly the same force but in opposite directions. Mathematically, we take our master equation and substitute what we just found:

Moving the term to the other side, we discover that the second wave, evaluated at $x=0$, must be exactly:

Now comes the fun part: playing detective with our options. We need to find the wave equation that gives us $-a\sin (\omega t)$ when we plug in $x=0$. Let's test option (a): $y=a\sin (\omega t+kx)$. If $x=0$, this becomes positive $a\sin (\omega t)$. If we added this to our first wave, the displacements would double, creating an antinode, not a node! So, option (a) is out.

Now let's look at option (b): $y=-a\sin (\omega t+kx)$. Plugging in $x=0$ gives us exactly $-a\sin (\omega t)$. This is a perfect match! It perfectly cancels the first wave at the origin. So, we have our winner. The unknown wave is:

But let's think about what this equation actually represents physically. The $+kx$ inside the sine function tells us that this wave is traveling in the negative $x$ direction. It's moving opposite to our original wave. And the negative sign out in front? That represents a phase shift of $\pi$ radians, or $180^{\circ }$. This is exactly what happens when a wave reflects off a rigid, fixed boundary. The wave flips upside down as it bounces back, ensuring that the point of attachment never moves.