In the resonance tube experment, the first and second states of resonance are observed at 20 cm and 670 cm. find the value of end correction (in cm).

Answer

To find the value of the end correction (E) in the resonance tube experiment, we can follow these steps:

### Step 1: Understand the Concept of End Correction
In the resonance tube experiment, the length of the air column (L) is not just the physical length of the tube filled with water. The sound wave resonates at certain lengths, and due to the nature of wave behavior, there is an additional length at the open end of the tube known as the end correction (E). This means that the effective length of the air column is L + E.

### Step 2: Set Up the Equations for Resonance
For the first state of resonance:
- The effective length is given by:
\[ L_1 + E = \frac{\lambda}{4} \quad \text{(Equation 1)} \]
For the second state of resonance:
- The effective length is given by:
\[ L_2 + E = \frac{3\lambda}{4} \quad \text{(Equation 2)} \]

### Step 3: Substitute the Given Values
From the problem, we have:
- \( L_1 = 20 \, \text{cm} \)
- \( L_2 = 670 \, \text{cm} \)

### Step 4: Rearrange the Equations
From Equation 1:
\[ E = \frac{\lambda}{4} - L_1 \]
From Equation 2:
\[ E = \frac{3\lambda}{4} - L_2 \]

### Step 5: Equate the Two Expressions for E
Setting the two expressions for E equal gives:
\[ \frac{\lambda}{4} - L_1 = \frac{3\lambda}{4} - L_2 \]

### Step 6: Solve for λ
Rearranging the equation:
\[ L_2 - L_1 = \frac{3\lambda}{4} - \frac{\lambda}{4} \]
\[ L_2 - L_1 = \frac{2\lambda}{4} = \frac{\lambda}{2} \]
Thus,
\[ \lambda = 2(L_2 - L_1) \]

### Step 7: Substitute L1 and L2
Substituting the values of \( L_1 \) and \( L_2 \):
\[ \lambda = 2(670 - 20) = 2(650) = 1300 \, \text{cm} \]

### Step 8: Substitute λ Back to Find E
Now, we can substitute \( \lambda \) back into one of the equations for E. Using Equation 1:
\[ L_1 + E = \frac{\lambda}{4} \]
\[ 20 + E = \frac{1300}{4} \]
\[ 20 + E = 325 \]
\[ E = 325 - 20 = 305 \, \text{cm} \]

### Final Answer
The value of the end correction (E) is:
\[ \boxed{305 \, \text{cm}} \]