This question describes the kinds of signals sentient beings would transmit to let others know of their existence. The methods described are geared towards communicating the mere existence of intelligent life. But what if an alien civilization wanted to send an SOS signal, asking for help? What kind of message would they transmit to have the highest chance of someone hearing and understanding it?
Really, there are three parts to your question:
- Transmitting the message
- Translating the message
- Translating the meaning
Number 1 is rather straightforward and there are tons of resources and other questions where you can read about sending messages at interstellar distances. Whether the signal is via radio, smoke signal, enormous flash light, or something else, making a signal receivable across interstellar distances is well within the capabilities of your standard dedicated multi-billionaire/small country today. A bigger problem would probably be that with current technology all messages would be limited to light speed and would probably end up arriving hundreds if not tens of thousands of years too late for anyone to reasonably help.
Number 2 is a bit more esoteric, but still possible in my opinion. Smarter people than I have worked at the problem of making a message decipherable to most conceivable alien intelligences. Usually, these approaches start with the understanding of a fundamental concept, such as math or elemental composition, and work upwards. I think it's reasonable to assume that through work and sufficient complexity, a language instruction system could be built which works its way up to something like English eventually.
Number 3 is a complete shot in the dark. Currently, we have no evidence of alien intelligences past or present. All we have is a couple hundred years of science-fiction writers and scientists taking their best (wild) guesses, but really, we have no clue at all what an alien might be like. Maybe, bipedal beings who navigate primarily by observing visual light are the standard, but maybe they're not. Additionally, there are meanings which may be impossible to convey to an alien mind. For example, take a core human emotion: jealousy. What if the aliens have no sense of self? No I? Yes, distress, fear, and asking for help all seem like rather fundamental animal drives and emotions, but as our sample size of "life" is one (earth), we can't really do more than speculate wildly on what aliens might be like.
If I had to do this my message would be (in English):
"Please send help, we're in trouble! Our problem is [...]"
And then I'd transmit the entirety of human knowledge (all literature, media, internet, etc) along with the message so that they can figure it out themselves.
They'd send an SOS
It would be a short repeating pattern to get attention. Needs to be complex enough to be not natural but simple enough to be short and blasted at high power that it will get noticed.
Any alien lifeforms capable of receiving such message won't understand the language but will know someone has sent an alert because it's not background noise and it's too short to be a message.
Chaos is danger. The disruption of order by chaos is a call for help
Intelligent life must rely on patterns and pattern recognition to obtain energy in opposition to the forces of entropy. Intelligence enables pattern recognition in the service of life. Unpredictability and the failure of patterns is dangerous to life.
A sequence of prime numbers is a fine way to convey intelligence - a pattern, and one that would be recognized only by an intelligence and not produced by a natural phenomenon (as something like the Fibonacci series might).
The distress call would be disruption of the sequence of primes by random numbers, which would end the sequence. The sequence of primes would then restart at the beginning and again be disrupted and ended at a different point by random numbers: the disruption of order by chaos, anathema to life.
Show, don't tell.
It's really unlikely that any detailed explanation of a problem could be transmitted and then translated in a reasonable amount of time. So I'll just focus on sending a generic "We have trouble" signal. If you are lucky, your neighbours would decide to help and then, with a 2-way contact, it would be certainly easier to explain the exact problem.
So, what could be a likely sign of intelligent life in trouble? I'd argue that a few high-altitude nuclear explosions would send the message. Maybe reducing radio transmissions afterwards, to make it clear the explosions caused some damage. Or starting rapidly changing atmosphere composition and albedo.
Coincidentally, humanity has sent such SOS signal already.
Send a constant like Pi or e in binary at the hydrogen line, continuously for at least several days, preferably as long as possible. Use a frequency of: $$x\pi\left(1420.40575177\right)\ MHz$$
x = any natural number
This will ensure that the receiving civilization is aware of the sending civilization’s intelligence, and it cannot be a naturally generated harmonic if it is at the frequency shown above.
If the receiving civilization continues to hear the message, they will pinpoint the source of the message. They can then try to contact the civilization out of curiosity, even if they don’t know that they need help.
Regardless of language, the simplest signal would be a repeating sequence message or something whose audio interpretation consists of a mathematical pattern; a pattern that doesn't exist in nature. 1 1-2 1-2-4 1-2-4-8 1 1-2 1-2-4 1-2-4-8 or a exponential regression 1 1-3 1-3-9 1-3-9-27 A distress signal cannot be interpreted as "Distress" unless one has prior knowledge of their signals capability to be used in such fashion.
The language of mathematics
The process for sending an SOS message would be the same as the process for sending any other type of message.
Lambros D. Callimahos of the NSA wrote a technical thought-experiment paper titled "Communication with Extraterrestrial Intelligence", which has now ben declassified. You can easily find it on Google.
It explains how the message should establish a mathematical vocabulary and grammar by starting with extremely simple math equations and then increasing in complexity.
This paper quotes British mathematician Lancelot Hogben, who I think put it perfectly: "Number is the most universal concept for establishing communication between intelligent beings; therefore, mathematics forms the basis for the first steps in extraterrestrial communication."
After this, physical values (such as the constants on the periodic table) could be used to define words (e.g. for all the periodic elements).
This form of defining new words continues, using previously defined words, until you have defined all of the vocabulary you need for your message.
After this, an SOS message (or any other message) could be crafted.