If the gain of a system at every particular delay is constant, the system will produce oscillations with those periods which have a gain of one. At periods where the gain exceeds one, the strength of the oscillations will grow unless or until something causes the gain falls to below one. If there were one frequency where the gain stabilized at exactly one, and it were less than one at all other frequencies, the circuit would produce a sine wave at the frequency in question. The wave would be a sine wave because any other type of wave would have content at frequencies where the gain is less than one.
Note that in practice, many types of oscillating circuits have gains which vary widely during the course of each "cycle". Such variations make it very difficult to predict analytically the frequency content of their output. Because there is a very fine line between having oscillations die down to nothing, and having oscillations grow without bound, even circuits which are intended to produce sine waves generally end up having a gain which is sometimes greater than 1 and sometimes less than one, though ideally there's a gain control mechanism that will try to settle on the right value.
Incidentally, some circuits use incandescent light bulbs for that purpose, since their resistance varies with temperature. If the power fed to a light bulb is proportional to the strength of an oscillator's signal, and if an increase in resistance will cause a reduction in gain, then the light bulb's temperature will tend to reach an equilibrium where the gain is 1. If the frequency in question is fast enough, the light bulb will only heat up or cool down a little bit during each cycle, allowing reasonably-clean sine waves to be generated.
Addendum
Rather than using the term "constant gain", it may be more helpful to use the term "linear circuit". To borrow an analogy from a magazine I read some years back, comparing "linear circuits" to "non-linear circuits" is like comparing "kangaroo biology" to "non-kangaroo biology"; linear circuits are a particular subcategory of circuits, and non-linear circuits are everything else.
A one-input one-output linear black box is one which takes an input signal and produces an output signal, with the characteristic that if F(x) represents the signal produced by the box when it is fed signal x, and if A and B are two input signals, then F(A+B) will equal F(A)+F(B). There are many kinds of things a linear black box can do to a signal, but all must obey the above criterion. The output produced by a linear black box when given a combination of many different frequency signals will be the sum of the outputs that would be produced for each frequency in isolation.
The behavior of many practical circuits is close to that of a one-input one-output linear black box. Since any wave other than a sine wave is a combination of sine waves at multiple frequencies, for a circuit to oscillate with anything other than a sine wave, there must be multiple frequencies which, if fed in individually, would cause the output to precisely match the input in phase and amplitude. While it is certainly possible to construct such circuits, most practical circuits will only have one frequency where such behavior will occur.
Then I used lower values of resistance, and for example, for
resistance of 10 Ohms on output, amplitude of Vout is about 0.13V.
A 741 is NOT a power amplifier - putting 10 ohms on the output is a recipe for disaster. Read the data sheet...
+/- 25mA is the output short circuit current and it's not recommended for anything like 10 ohms. In the data sheet it talks about the minimum load being 2 kohm to meet other technical values.
What are applications of Wien oscillator?
The process of learning to read data sheets and understanding op-amps will expose you to many applications and if you think that's an unfair answer then "tough" because questions that solicit opinions are usually closed down.
Best Answer
I think, in principle your analysis is correct. And - yes, the resistors R15 and R16 provide a DC reference which enables a dc operating point (output voltage) at app. half of the powwer supply. Therefore - in contrast to your assumption, the output will consist of a DC voltage and a superimposed full sine wave.
The frequency of the sine wave wo (provided the diodes allow a good amplitude control) will be - of course - in accordance with the Barkhausen criterion. In this context, you have nothing to do than to find the mid frequency wo (center frequency) of the RC-bandpass in the positive feddback path. At w=wo the phase shift of this bandpass will be zero and the feedback factor 1/3.
Comment: The above mentioned facor (1/3) applies only if the bandpass in the positive feedback path is symmetric (equal R and equal C in the series resp. the parallel RC combination). In your case, there is an additional 10k pot (R20 ?) in series with the 30k resistor. This resistor slightly detunes the bandpass and, thus, can vary the oscillator frequency within a certain tuning range. As a consequency, the feedback factor (nominal 1/3) will slighly change as well. This will be, however, not a problem as long as the amplitude regulation mechanism can cope for this variation.