Statement-1`:` Two longitudinal waves given by equation `y_(1)(x,t)=2asin(omegat-kx)`
and `y_(2)(x,t)=a sin (2 omegat-2kx)`
will have equal intensity.
Statement-2 `:` Intensity of waves of given frequency in same medium is proportional to square of amplitude only.

Options

A. (a)Statement I is false, Statement II is true
B. (b)Statement I is true, Statement II is false
C. (c)Statement I is true, Statement II is true, Statement II is the correct explanation of Statement I
D. (d)Statement I is true, Statement II is true, Statement II is not correct explanation of Statement I

Explanation

To analyze the given statements, we need to evaluate the intensity of the two longitudinal waves described by the equations:

1. \( y_1(x,t) = 2a \sin(\omega t - kx) \)
2. \( y_2(x,t) = a \sin(2\omega t - 2kx) \)

### Step 1: Calculate the intensity of the first wave \( y_1 \)

The intensity \( I \) of a wave is given by the formula:

\[ I = \frac{1}{2} \rho v \omega^2 A^2 \]

where:
- \( \rho \) is the density of the medium,
- \( v \) is the velocity of the wave,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the wave.

For the first wave \( y_1 \):
- Amplitude \( A_1 = 2a \)
- Angular frequency \( \omega_1 = \omega \)

Substituting these values into the intensity formula:

\[ I_1 = \frac{1}{2} \rho v \omega^2 (2a)^2 = \frac{1}{2} \rho v \omega^2 \cdot 4a^2 = 2 \rho v \omega^2 a^2 \]

### Step 2: Calculate the intensity of the second wave \( y_2 \)

For the second wave \( y_2 \):
- Amplitude \( A_2 = a \)
- Angular frequency \( \omega_2 = 2\omega \)

Substituting these values into the intensity formula:

\[ I_2 = \frac{1}{2} \rho v (2\omega)^2 (a)^2 = \frac{1}{2} \rho v \cdot 4\omega^2 \cdot a^2 = 2 \rho v \omega^2 a^2 \]

### Step 3: Compare the intensities \( I_1 \) and \( I_2 \)

From the calculations, we see that:

\[ I_1 = 2 \rho v \omega^2 a^2 \]
\[ I_2 = 2 \rho v \omega^2 a^2 \]

Thus, \( I_1 = I_2 \), which means the first statement is true: the two waves have equal intensity.

### Step 4: Evaluate Statement 2

Statement 2 claims that the intensity of waves of a given frequency in the same medium is proportional to the square of the amplitude only. This is indeed true, as we derived the intensity formula, which shows that intensity is proportional to \( A^2 \) when frequency and medium properties are constant.

### Conclusion

- **Statement 1** is true.
- **Statement 2** is true.
- However, Statement 2 does not provide a correct explanation for Statement 1 since it does not account for the angular frequency differences.

### Final Answer

Both statements are true, but Statement 2 is not a correct explanation for Statement 1.