Wave Equation Construction

1. The Recipe for a Wave Equation

Constructing a wave equation is like following a recipe. You need to gather a few key ingredients (the physical parameters) and combine them in the correct way.

To get a complete picture of the process, please watch the following video from Khan Academy. It masterfully builds the equation from the ground up, starting with a snapshot in space and then setting it in motion through time.

As you saw in the video, to write the equation for a specific wave, you need to determine the following five components:

1. Amplitude ($A$): This is the maximum displacement from equilibrium. It's the number that multiplies the sine or cosine function.
2. Waveform (sin or cos): This is determined by the wave's displacement at the origin ($x=0$) at the start time ($t=0$).
   • If $y(0,0)=0$ and the wave goes up initially, use sin.
   • If $y(0,0)=A$ (a crest), use cos.
   • If the starting condition is different, you'll need an initial phase ($\varphi$), which we will discuss more in complex problems. For most standard cases, you can start with sine or cosine.
3. Angular Wave Number ($k$): You'll almost always be given the wavelength $\lambda$. Calculate $k$ using $k=2\pi /\lambda$.
4. Angular Frequency ($\omega$): You'll be given either the frequency $f$ or the period $T$. Calculate $\omega$ using $\omega =2\pi f$ or $\omega =2\pi /T$.
5. Direction of Travel: This determines the sign inside the brackets. Remember, for the form $(kx\pm \omega t)$:
   • Positive x-direction $⟹$ use a minus sign: $(kx-\omega t)$.
   • Negative x-direction $⟹$ use a plus sign: $(kx+\omega t)$.

Once you have these five pieces, you assemble them into the final equation.

2. Practice: Constructing an Equation

Let's apply this recipe with a typical NEET-style problem.

Problem: A transverse wave on a string is traveling in the positive x-direction. It has an amplitude of 10 cm, a wavelength of 80 cm, and a frequency of 12 Hz. The displacement at $x=0$ and $t=0$ is zero. Write the equation describing this wave, using SI units.

Solution:

1. Convert to SI Units: This is a crucial first step.

   • Amplitude: $A=10\text{ cm}=0.1\text{ m}$
   • Wavelength: $\lambda =80\text{ cm}=0.8\text{ m}$
   • Frequency: $f=12\text{ Hz}$ (already in SI units)

2. Determine Parameters:

   • Amplitude (A): $A=0.1\text{ m}$
   • Waveform: Since $y(0,0)=0$, we will use a sine function. This implies the initial phase $\varphi =0$.
   • Angular Wave Number (k):
     $k=\frac{2\pi }{\lambda }=\frac{2\pi }{0.8}=\frac{20\pi }{8}=2.5\pi \text{ rad/m}$
   • Angular Frequency ($\omega$):
     $\omega =2\pi f=2\pi (12)=24\pi \text{ rad/s}$
   • Direction: The wave travels in the positive x-direction, so we use a minus sign.

3. Assemble the Equation: Now, we plug these values into the standard form $y(x,t)=A\sin (kx-\omega t)$.
   $y(x,t)=0.1\sin (2.5\pi x-24\pi t)$

This is the final equation describing the wave in SI units.

3. The Reverse Process: Deconstructing an Equation

Just as important as building the equation is being able to read one. In many problems, you'll be given the equation and asked to find the wave's properties. The process is simply a matter of comparison.

Your NCERT textbook provides an excellent guide and a worked example for this.

Let's summarize the strategy from the NCERT example:

Problem: A wave is described by $y(x,t)=0.005\sin (80.0x-3.0t)$, where all values are in SI units. Find its amplitude, wavelength, period, and frequency.

Solution Strategy:

1. Write the Standard Form: Write the general equation below the given one.
   Given: $y(x,t)=0.005\sin (80.0x-3.0t)$
   Standard: $y(x,t)=A\sin (kx-\omega t)$

2. Compare and Identify: By direct comparison, you can see:

   • $A=0.005\text{ m}$
   • $k=80.0\text{ rad/m}$
   • $\omega =3.0\text{ rad/s}$

3. Calculate Physical Properties: Now use the definitions of $k$ and $\omega$ to find $\lambda$, $T$, and $f$.

   • Wavelength ($\lambda$): Since $k=2\pi /\lambda$, we have:
     $\lambda =\frac{2\pi }{k}=\frac{2\pi }{80.0}\approx 0.0785\text{ m}\text{ or }7.85\text{ cm}$
   • Period ($T$): Since $\omega =2\pi /T$, we have:
     $T=\frac{2\pi }{\omega }=\frac{2\pi }{3.0}\approx 2.09\text{ s}$
   • Frequency ($f$):
     $f=\frac{1}{T}=\frac{1}{2.09}\approx 0.48\text{ Hz}$

This methodical comparison is your most reliable tool for these problems.

4. A Key Relationship: Wave Speed

Finally, let's briefly touch upon a concept that ties everything together and previews our next lesson: wave speed ($v$).

You already know the fundamental relationship:
$v=f\lambda$

There is also a very useful shortcut for finding the wave speed directly from the equation's parameters, $\omega$ and $k$:
$v=\frac{\omega }{k}$
You can derive this easily: $v=f\lambda =(\omega /2\pi )\times (2\pi /k)=\omega /k$. This formula is a massive time-saver in exams.

For the wave in our last example, $y(x,t)=0.005\sin (80.0x-3.0t)$, the wave speed is:
$v=\frac{\omega }{k}=\frac{3.0\text{ rad/s}}{80.0\text{ rad/m}}=0.0375\text{ m/s}$

Conclusion

Today, you've learned the practical craft of translating between a wave's physical description and its mathematical one. This two-way skill is essential for solving a wide range of physics problems.

Key Takeaways:

• To construct a wave equation: You need its amplitude, wavelength, frequency, direction, and initial state (which determines the sine/cosine form).
• The recipe: Calculate $k=2\pi /\lambda$ and $\omega =2\pi f$, determine the sign for direction, and assemble the equation $y(x,t)=A\sin (kx\pm \omega t+\varphi )$.
• To deconstruct a wave equation: Compare it directly to the standard form to identify $A$, $k$, and $\omega$. From these, you can calculate all other properties like $\lambda$, $f$, $T$, and $v$.
• Wave speed shortcut: $v=\omega /k$ is a quick and powerful tool.

In our next lesson, we will focus more deeply on the relationship between wave speed, frequency, and wavelength ($v=f\lambda$) and solve a variety of problems related to it.