The amplitude of a wave represented by displacement equation `y=(1)/(sqrta)sinomegat+-(1)/(sqrtb)cosomegat` will be

Options

A. `(a+b)/(ab)`
B. `(sqrta+sqrtb)/(ab)`
C. `(sqrta+-sqrtb)/(ab)`
D. `sqrt((a+b)/(ab))`

Explanation

To find the amplitude of the wave represented by the displacement equation \( y = \frac{1}{\sqrt{a}} \sin(\omega t) \pm \frac{1}{\sqrt{b}} \cos(\omega t) \), we can follow these steps:

### Step 1: Rewrite the equation
The displacement equation can be expressed as:
\[ y = \frac{1}{\sqrt{a}} \sin(\omega t) + \left(\pm \frac{1}{\sqrt{b}} \cos(\omega t)\right) \]

### Step 2: Convert cosine to sine
We can rewrite the cosine term in terms of sine:
\[ \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right) \]
Thus, we can express the equation as:
\[ y = \frac{1}{\sqrt{a}} \sin(\omega t) \pm \frac{1}{\sqrt{b}} \sin\left(\omega t + \frac{\pi}{2}\right) \]

### Step 3: Identify the amplitudes
In this equation, we have two components:
- The amplitude of the sine term is \( A_1 = \frac{1}{\sqrt{a}} \)
- The amplitude of the cosine term (now in sine form) is \( A_2 = \frac{1}{\sqrt{b}} \)

### Step 4: Use the amplitude formula
The formula for the resultant amplitude \( A \) when combining two waves with a phase difference \( \theta \) is:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\theta)} \]
In our case, the phase difference \( \theta \) is \( \frac{\pi}{2} \), and \( \cos\left(\frac{\pi}{2}\right) = 0 \). Therefore, the formula simplifies to:
\[ A = \sqrt{A_1^2 + A_2^2} \]

### Step 5: Substitute the values
Substituting the values of \( A_1 \) and \( A_2 \):
\[ A = \sqrt{\left(\frac{1}{\sqrt{a}}\right)^2 + \left(\frac{1}{\sqrt{b}}\right)^2} \]
\[ A = \sqrt{\frac{1}{a} + \frac{1}{b}} \]

### Step 6: Final expression
Thus, the amplitude of the wave is:
\[ A = \sqrt{\frac{b + a}{ab}} \]

### Conclusion
The amplitude of the wave represented by the given displacement equation is \( A = \sqrt{\frac{b + a}{ab}} \).

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