Which of the following results in more energy in a wave?

Options

A. a smaller wavelength
B. a lower frequecy
C. a shallower amplitude
D. a lower speed

Explanation

To determine which factor results in more energy in a wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Formula**:
The energy \( E \) of a wave is given by the formula:
\[ E = h \cdot \nu \]
where \( h \) is Planck's constant and \( \nu \) (nu) is the frequency of the wave.

2. **Analyze the Relationship**:
From the formula, we can see that energy is directly proportional to frequency. This means that if the frequency increases, the energy of the wave also increases.

3. **Consider Wavelength**:
The relationship between frequency \( \nu \) and wavelength \( \lambda \) is given by the equation:
\[ \nu \cdot \lambda = c \]
where \( c \) is the speed of light. This implies that if the wavelength decreases, the frequency must increase to keep the speed constant.

4. **Determine the Effect of Wavelength on Energy**:
Since energy is directly related to frequency, and frequency is inversely related to wavelength, reducing the wavelength will result in an increase in energy. Therefore, a smaller wavelength corresponds to more energy.

5. **Evaluate Other Factors**:
- **Amplitude**: The amplitude of a wave affects its intensity but not its energy directly. Higher amplitude means higher intensity, but it does not imply more energy in the context of the wave's frequency.
- **Speed**: Slowing down the speed of the wave does not affect the frequency directly; it affects the wavelength. If frequency remains constant, energy also remains constant.

6. **Conclusion**:
The option that results in more energy in a wave is a smaller wavelength.

### Final Answer:
A smaller wavelength results in more energy in a wave.

---